Divisibility Rules and Factors Competitive Exams
100 questions · Mathematics
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Question 1 of 100
If the four-digit number 5x34 is divisible by 11, find the digit x.
ExplanationFor divisibility by 11, (sum of digits at odd places) - (sum of digits at even places) must be 0 or a multiple of 11. In 5x34: odd places = 5 + 3 = 8; even places = x + 4. Difference = 8 - (x + 4) = 4 - x, which is 0 or a multiple of 11 only for x = 4. Check: 5434 = 11 × 494.
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Question 2 of 100
The HCF and LCM of two numbers are 4 and 48. If one number is 12, find the other.
ExplanationProduct of two numbers = HCF × LCM. So, 12 × x = 4 × 48 => x = 16. Check: HCF(12, 16) = 4 and LCM(12, 16) = 48.
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Question 3 of 100
Which of the following rules confirms 1016 is divisible by 2?
ExplanationA number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. For 1016, the last digit is even.
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Question 4 of 100
What is the LCM of 6, 10, and 2?
ExplanationThe smallest number divisible by 6, 10, and 2 is 30.
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Question 5 of 100
Which is the divisibility rule for 4?
ExplanationRule for 4: the number formed by the last two digits must be divisible by 4 (because 100 is divisible by 4). Example: 3,716 is divisible by 4 since 16 is.
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Question 6 of 100
Find the greatest number that divides both 16 and 16.
ExplanationGreatest number dividing both is their HCF, which is 16.
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Question 7 of 100
Find the Least Common Multiple (LCM) of 9 and 13.
ExplanationLCM = (a * b) / HCF. Here LCM is 117. Tip: LCM is always ≥ the largest number.
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Question 8 of 100
What is the Highest Common Factor (HCF) of 30 and 21?
ExplanationThe factors common to both are checked; the highest is 3. Tip: HCF is always ≤ the smallest number.
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Question 9 of 100
Which of the following rules confirms 1674 is divisible by 2?
ExplanationA number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. For 1674, the last digit is even.
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Question 10 of 100
Find the Least Common Multiple (LCM) of 8 and 14.
ExplanationLCM = (a * b) / HCF. HCF(8, 14) = 2, so LCM = (8 × 14) / 2 = 56. Tip: LCM is always ≥ the largest number.
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Question 11 of 100
The HCF and LCM of two numbers are 15 and 30. If one number is 15, find the other.
ExplanationProduct of two numbers = HCF × LCM. So, 15 × x = 15 × 30 => x = 30.
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Question 12 of 100
Find the Least Common Multiple (LCM) of 10 and 17.
ExplanationLCM = (a * b) / HCF. Here LCM is 170. Tip: LCM is always ≥ the largest number.
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Question 13 of 100
Which is the divisibility rule for 8?
ExplanationRule for 8: Last three digits divide by 8. Memorizing divisibility rules saves calculation time.
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Question 14 of 100
Find the greatest number that divides both 21 and 22.
ExplanationGreatest number dividing both is their HCF, which is 1.
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Question 15 of 100
If the four-digit number 6x25 is divisible by 11, find the digit x.
ExplanationFor divisibility by 11, (sum of digits at odd places) - (sum of digits at even places) must be 0 or a multiple of 11. In 6x25: odd places = 6 + 2 = 8; even places = x + 5. Difference = 8 - (x + 5) = 3 - x, which is 0 or a multiple of 11 only for x = 3. Check: 6325 = 11 × 575.
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Question 16 of 100
What is the Highest Common Factor (HCF) of 14 and 29?
ExplanationThe factors common to both are checked; the highest is 1. Tip: HCF is always ≤ the smallest number.
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Question 17 of 100
Which of the following rules confirms 1302 is divisible by 2?
ExplanationA number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. For 1302, the last digit is even.
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Question 18 of 100
The HCF and LCM of two numbers are 3 and 42. If one number is 6, find the other.
ExplanationProduct of two numbers = HCF × LCM. So, 6 × x = 3 × 42 => x = 21. Check: HCF(6, 21) = 3 and LCM(6, 21) = 42.
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Question 19 of 100
If the four-digit number 4x27 is divisible by 9, find the digit x.
ExplanationFor divisibility by 9, the sum of digits must be a multiple of 9. In 4x27: 4+2+7 + x = 13 + x, which is a multiple of 9 only for x = 5. Check: 4527 = 9 × 503.
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Question 20 of 100
What is the LCM of 6, 3, and 10?
ExplanationThe smallest number divisible by 6, 3, and 10 is 30.
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Question 21 of 100
Which is the divisibility rule for 5?
ExplanationRule for 5: the last digit must be 0 or 5. Example: 1,240 and 1,245 are both divisible by 5.
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Question 22 of 100
What is the Highest Common Factor (HCF) of 25 and 33?
ExplanationThe factors common to both are checked; the highest is 1. Tip: HCF is always ≤ the smallest number.
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Question 23 of 100
Find the Least Common Multiple (LCM) of 10 and 18.
ExplanationLCM = (a * b) / HCF. Here LCM is 90. Tip: LCM is always ≥ the largest number.
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Question 24 of 100
Why is 738 divisible by 3?
ExplanationDivisibility rule of 3: sum of digits must be a multiple of 3. Tip: Add digits until you get a single digit to check.
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Question 25 of 100
Which of the following rules confirms 1564 is divisible by 2?
ExplanationA number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. For 1564, the last digit is even.
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Question 26 of 100
Why is 381 divisible by 3?
ExplanationDivisibility rule of 3: sum of digits must be a multiple of 3. Tip: Add digits until you get a single digit to check.
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Question 27 of 100
How many total factors does 48 have?
ExplanationFactors of 48 are [1, 2, 3, 4, 6, 8, 12, 16, 24, 48]. Total count is 10. Tip: Prime factorize n = p^a * q^b, total factors = (a+1)(b+1).
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Question 28 of 100
Find the greatest number that divides both 28 and 30.
ExplanationGreatest number dividing both is their HCF, which is 2.
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Question 29 of 100
How many total factors does 44 have?
ExplanationFactors of 44 are [1, 2, 4, 11, 22, 44]. Total count is 6. Tip: Prime factorize n = p^a * q^b, total factors = (a+1)(b+1).
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Question 30 of 100
Which is the divisibility rule for 3?
ExplanationRule for 3: Sum of digits is multiple of 3. Memorizing divisibility rules saves calculation time.
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Question 31 of 100
The HCF and LCM of two numbers are 10 and 20. If one number is 10, find the other.
ExplanationProduct of two numbers = HCF × LCM. So, 10 × x = 10 × 20 => x = 20.
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Question 32 of 100
Find the greatest number that divides both 11 and 28.
ExplanationGreatest number dividing both is their HCF, which is 1.
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Question 33 of 100
Why is 807 divisible by 3?
ExplanationDivisibility rule of 3: sum of digits must be a multiple of 3. Tip: Add digits until you get a single digit to check.
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Question 34 of 100
What is the LCM of 3, 8, and 9?
ExplanationThe smallest number divisible by 3, 8, and 9 is 72.
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Question 35 of 100
Which of the following rules confirms 1904 is divisible by 2?
ExplanationA number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. For 1904, the last digit is even.
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Question 36 of 100
The HCF and LCM of two numbers are 5 and 60. If one number is 15, find the other.
ExplanationProduct of two numbers = HCF × LCM. So, 15 × x = 5 × 60 => x = 20. Check: HCF(15, 20) = 5 and LCM(15, 20) = 60.
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Question 37 of 100
How many total factors does 37 have?
ExplanationFactors of 37 are [1, 37]. Total count is 2. Tip: Prime factorize n = p^a * q^b, total factors = (a+1)(b+1).
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Question 38 of 100
Why is 783 divisible by 3?
ExplanationDivisibility rule of 3: sum of digits must be a multiple of 3. Tip: Add digits until you get a single digit to check.
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Question 39 of 100
What is the Highest Common Factor (HCF) of 17 and 40?
ExplanationThe factors common to both are checked; the highest is 1. Tip: HCF is always ≤ the smallest number.
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Question 40 of 100
Find the Least Common Multiple (LCM) of 9 and 19.
ExplanationLCM = (a * b) / HCF. Here LCM is 171. Tip: LCM is always ≥ the largest number.
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Question 41 of 100
What is the Highest Common Factor (HCF) of 29 and 40?
ExplanationThe factors common to both are checked; the highest is 1. Tip: HCF is always ≤ the smallest number.
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Question 42 of 100
Find the Least Common Multiple (LCM) of 7 and 15.
ExplanationLCM = (a * b) / HCF. Here LCM is 105. Tip: LCM is always ≥ the largest number.
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Question 43 of 100
How many total factors does 27 have?
ExplanationFactors of 27 are [1, 3, 9, 27]. Total count is 4. Tip: Prime factorize n = p^a * q^b, total factors = (a+1)(b+1).
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Question 44 of 100
Find the Least Common Multiple (LCM) of 7 and 18.
ExplanationLCM = (a * b) / HCF. Here LCM is 126. Tip: LCM is always ≥ the largest number.
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Question 45 of 100
The HCF and LCM of two numbers are 9 and 18. If one number is 9, find the other.
ExplanationProduct of two numbers = HCF × LCM. So, 9 × x = 9 × 18 => x = 18.
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Question 46 of 100
Why is 777 divisible by 3?
ExplanationDivisibility rule of 3: sum of digits must be a multiple of 3. Tip: Add digits until you get a single digit to check.
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Question 47 of 100
What is the LCM of 10, 6, and 7?
ExplanationThe smallest number divisible by 10, 6, and 7 is 210.
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Question 48 of 100
Find the greatest number that divides both 26 and 17.
ExplanationGreatest number dividing both is their HCF, which is 1.
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Question 49 of 100
Why is 384 divisible by 3?
ExplanationDivisibility rule of 3: sum of digits must be a multiple of 3. Tip: Add digits until you get a single digit to check.
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Question 50 of 100
If the four-digit number 9x25 is divisible by 9, find the digit x.
ExplanationFor divisibility by 9, the sum of digits must be a multiple of 9. In 9x25: 9+2+5 + x = 16 + x, which is a multiple of 9 only for x = 2. Check: 9225 = 9 × 1025.
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Question 51 of 100
How many total factors does 45 have?
ExplanationFactors of 45 are [1, 3, 5, 9, 15, 45]. Total count is 6. Tip: Prime factorize n = p^a * q^b, total factors = (a+1)(b+1).
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Question 52 of 100
Find the greatest number that divides both 26 and 23.
ExplanationGreatest number dividing both is their HCF, which is 1.
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Question 53 of 100
What is the Highest Common Factor (HCF) of 30 and 20?
ExplanationThe factors common to both are checked; the highest is 10. Tip: HCF is always ≤ the smallest number.
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Question 54 of 100
Why is 780 divisible by 3?
ExplanationDivisibility rule of 3: sum of digits must be a multiple of 3. Tip: Add digits until you get a single digit to check.
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Question 55 of 100
If the four-digit number 8x19 is divisible by 11, find the digit x.
ExplanationFor divisibility by 11, (sum of digits at odd places) - (sum of digits at even places) must be 0 or a multiple of 11. In 8x19: odd places = 8 + 1 = 9; even places = x + 9. Difference = 9 - (x + 9) = 0 - x, which is 0 or a multiple of 11 only for x = 0. Check: 8019 = 11 × 729.
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Question 56 of 100
The HCF and LCM of two numbers are 6 and 12. If one number is 6, find the other.
ExplanationProduct of two numbers = HCF × LCM. So, 6 × x = 6 × 12 => x = 12.
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Question 57 of 100
The HCF and LCM of two numbers are 12 and 24. If one number is 12, find the other.
ExplanationProduct of two numbers = HCF × LCM. So, 12 × x = 12 × 24 => x = 24.
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Question 58 of 100
What is the Highest Common Factor (HCF) of 19 and 16?
ExplanationThe factors common to both are checked; the highest is 1. Tip: HCF is always ≤ the smallest number.
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Question 59 of 100
How many total factors does 34 have?
ExplanationFactors of 34 are [1, 2, 17, 34]. Total count is 4. Tip: Prime factorize n = p^a * q^b, total factors = (a+1)(b+1).
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Question 60 of 100
If the four-digit number 2x58 is divisible by 9, find the digit x.
ExplanationFor divisibility by 9, the sum of digits must be a multiple of 9. In 2x58: 2+5+8 + x = 15 + x, which is a multiple of 9 only for x = 3. Check: 2358 = 9 × 262.
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Question 61 of 100
Find the Least Common Multiple (LCM) of 12 and 23.
ExplanationLCM = (a * b) / HCF. Here LCM is 276. Tip: LCM is always ≥ the largest number.
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Question 62 of 100
Which is the divisibility rule for 9?
ExplanationRule for 9: Sum of digits is multiple of 9. Memorizing divisibility rules saves calculation time.
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Question 63 of 100
How many total factors does 32 have?
ExplanationFactors of 32 are [1, 2, 4, 8, 16, 32]. Total count is 6. Tip: Prime factorize n = p^a * q^b, total factors = (a+1)(b+1).
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Question 64 of 100
Which is the divisibility rule for 11?
ExplanationRule for 11: find (sum of digits at odd places) - (sum of digits at even places); it must be 0 or a multiple of 11. Example: 2,728 gives (2+2) - (7+8) = -11, so 2,728 is divisible by 11.
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Question 65 of 100
Find the Least Common Multiple (LCM) of 12 and 13.
ExplanationLCM = (a * b) / HCF. Here LCM is 156. Tip: LCM is always ≥ the largest number.
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Question 66 of 100
Which is the divisibility rule for 2?
ExplanationRule for 2: Last digit even. Memorizing divisibility rules saves calculation time.
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Question 67 of 100
Find the Least Common Multiple (LCM) of 6 and 15.
ExplanationLCM = (a * b) / HCF. Here LCM is 30. Tip: LCM is always ≥ the largest number.
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Question 68 of 100
What is the LCM of 10, 7, and 7?
ExplanationThe smallest number divisible by 10, 7, and 7 is 70.
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Question 69 of 100
What is the LCM of 4, 5, and 10?
ExplanationThe smallest number divisible by 4, 5, and 10 is 20.
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Question 70 of 100
If the four-digit number 7x14 is divisible by 9, find the digit x.
ExplanationFor divisibility by 9, the sum of digits must be a multiple of 9. In 7x14: 7+1+4 + x = 12 + x, which is a multiple of 9 only for x = 6. Check: 7614 = 9 × 846.
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Question 71 of 100
Which is the divisibility rule for 6?
ExplanationRule for 6: Divisible by both 2 and 3. Memorizing divisibility rules saves calculation time.
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Question 72 of 100
Which is the divisibility rule for 10?
ExplanationRule for 10: the number must end in 0. Example: 3,450 is divisible by 10.
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Question 73 of 100
If the four-digit number 3x62 is divisible by 11, find the digit x.
ExplanationFor divisibility by 11, (sum of digits at odd places) - (sum of digits at even places) must be 0 or a multiple of 11. In 3x62: odd places = 3 + 6 = 9; even places = x + 2. Difference = 9 - (x + 2) = 7 - x, which is 0 or a multiple of 11 only for x = 7. Check: 3762 = 11 × 342.
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Question 74 of 100
Find the greatest number that divides both 26 and 25.
ExplanationGreatest number dividing both is their HCF, which is 1.
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Question 75 of 100
What is the Highest Common Factor (HCF) of 22 and 40?
ExplanationThe factors common to both are checked; the highest is 2. Tip: HCF is always ≤ the smallest number.
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Question 76 of 100
What is the Highest Common Factor (HCF) of 23 and 22?
ExplanationThe factors common to both are checked; the highest is 1. Tip: HCF is always ≤ the smallest number.
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Question 77 of 100
Which is the divisibility rule for 12?
ExplanationRule for 12: a number is divisible by 12 if it is divisible by both 3 and 4 (co-prime factors). Divisible by 2 and 3 only gives 6.
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Question 78 of 100
Which of the following rules confirms 712 is divisible by 2?
ExplanationA number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. For 712, the last digit is even.
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Question 79 of 100
What is the LCM of 7, 3, and 5?
ExplanationThe smallest number divisible by 7, 3, and 5 is 105.
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Question 80 of 100
Find the greatest number that divides both 11 and 13.
ExplanationGreatest number dividing both is their HCF, which is 1.
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Question 81 of 100
The HCF and LCM of two numbers are 16 and 32. If one number is 16, find the other.
ExplanationProduct of two numbers = HCF × LCM. So, 16 × x = 16 × 32 => x = 32.
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Question 82 of 100
What is the LCM of 2, 2, and 8?
ExplanationThe smallest number divisible by 2, 2, and 8 is 8.
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Question 83 of 100
Find the greatest number that divides both 22 and 16.
ExplanationGreatest number dividing both is their HCF, which is 2.
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Question 84 of 100
Which of the following rules confirms 1168 is divisible by 2?
ExplanationA number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. For 1168, the last digit is even.
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Question 85 of 100
What is the LCM of 5, 10, and 10?
ExplanationThe smallest number divisible by 5, 10, and 10 is 10.
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Question 86 of 100
What is the Highest Common Factor (HCF) of 21 and 18?
ExplanationThe factors common to both are checked; the highest is 3. Tip: HCF is always ≤ the smallest number.
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Question 87 of 100
If the four-digit number 5x63 is divisible by 9, find the digit x.
ExplanationFor divisibility by 9, the sum of digits must be a multiple of 9. In 5x63: 5+6+3 + x = 14 + x, which is a multiple of 9 only for x = 4. Check: 5463 = 9 × 607.
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Question 88 of 100
Why is 417 divisible by 3?
ExplanationDivisibility rule of 3: sum of digits must be a multiple of 3. Tip: Add digits until you get a single digit to check.
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Question 89 of 100
Find the greatest number that divides both 20 and 25.
ExplanationGreatest number dividing both is their HCF, which is 5.
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Question 90 of 100
How many total factors does 36 have?
ExplanationFactors of 36 are [1, 2, 3, 4, 6, 9, 12, 18, 36]. Total count is 9. Tip: Prime factorize n = p^a * q^b, total factors = (a+1)(b+1).
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Question 91 of 100
Why is 768 divisible by 3?
ExplanationDivisibility rule of 3: sum of digits must be a multiple of 3. Tip: Add digits until you get a single digit to check.
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Question 92 of 100
What is the LCM of 5, 8, and 5?
ExplanationThe smallest number divisible by 5, 8, and 5 is 40.
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Question 93 of 100
Which of the following rules confirms 898 is divisible by 2?
ExplanationA number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. For 898, the last digit is even.
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Question 94 of 100
How many total factors does 39 have?
ExplanationFactors of 39 are [1, 3, 13, 39]. Total count is 4. Tip: Prime factorize n = p^a * q^b, total factors = (a+1)(b+1).
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Question 95 of 100
How many total factors does 49 have?
ExplanationFactors of 49 are [1, 7, 49]. Total count is 3. Tip: Prime factorize n = p^a * q^b, total factors = (a+1)(b+1).
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Question 96 of 100
If the four-digit number 7x81 is divisible by 11, find the digit x.
ExplanationFor divisibility by 11, (sum of digits at odd places) - (sum of digits at even places) must be 0 or a multiple of 11. In 7x81: odd places = 7 + 8 = 15; even places = x + 1. Difference = 15 - (x + 1) = 14 - x, which is 0 or a multiple of 11 only for x = 3. Check: 7381 = 11 × 671.
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Question 97 of 100
Which of the following rules confirms 1964 is divisible by 2?
ExplanationA number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. For 1964, the last digit is even.
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Question 98 of 100
Which of the following rules confirms 1404 is divisible by 2?
ExplanationA number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. For 1404, the last digit is even.
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Question 99 of 100
The HCF and LCM of two numbers are 6 and 84. If one number is 12, find the other.
ExplanationProduct of two numbers = HCF × LCM. So, 12 × x = 6 × 84 => x = 42. Check: HCF(12, 42) = 6 and LCM(12, 42) = 84.
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Question 100 of 100
Why is 180 divisible by 3?
ExplanationDivisibility rule of 3: sum of digits must be a multiple of 3. Tip: Add digits until you get a single digit to check.
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