Question 1

What is the smallest 4-digit number that is divisible by both 8 and 12?

Accepted Answer

Step 1: Find LCM(8, 12). Prime factorization: 8 = 2³, 12 = 2² × 3. Step 2: LCM = 2³ × 3 = 24. Step 3: The smallest 4-digit number is 1000. Step 4: Divide 1000 by 24: 1000 ÷ 24 = 41.666... Step 5: Round up to 42. Step 6: Multiply: 42 × 24 = 1008. Step 7: Verify: 1008 ÷ 8 = 126 ✓, 1008 ÷ 12 = 84 ✓.

Question 2

Which of the following numbers is divisible by 11?

Accepted Answer

Step 1: Apply divisibility rule for 11: alternating sum of digits (starting from right) must be divisible by 11. Step 2: For 5742: (2 - 4 + 7 - 5) = 0, which is divisible by 11. Step 3: Therefore, 5742 is divisible by 11.

Question 3

A number is divisible by both 4 and 6. Which of the following must also divide this number?

Accepted Answer

Step 1: A number divisible by both 4 and 6 must be divisible by their LCM. Step 2: LCM(4, 6) = 12 (since 4 = 2², 6 = 2 × 3, LCM = 2² × 3 = 12). Step 3: Therefore, any number divisible by both 4 and 6 must be divisible by 12.

Question 4

A number leaves remainder 2 when divided by 5. Which of the following could be the last digit of this number?

Accepted Answer

Step 1: If a number leaves remainder 2 when divided by 5, the last digit determines divisibility by 5. Step 2: Numbers ending in 0 or 5 are divisible by 5 (remainder 0). Numbers ending in 1 or 6 leave remainder 1. Numbers ending in 2 or 7 leave remainder 2. Numbers ending in 3 or 8 leave remainder 3. Numbers ending in 4 or 9 leave remainder 4. Step 3: Therefore, the last digit must be 2 or 7.

Question 5

Which of the following numbers is NOT divisible by 3?

Accepted Answer

Step 1: Apply divisibility rule for 3: sum of all digits must be divisible by 3. Step 2: Check 4827: 4 + 8 + 2 + 7 = 21, divisible by 3. Check 5634: 5 + 6 + 3 + 4 = 18, divisible by 3. Check 7249: 7 + 2 + 4 + 9 = 22, NOT divisible by 3. Check 6351: 6 + 3 + 5 + 1 = 15, divisible by 3. Step 3: Therefore, 7249 is NOT divisible by 3.

Question 6

A number is divisible by 9. Which of the following statements must be true?

Accepted Answer

Step 1: Divisibility rule for 9: sum of all digits must be divisible by 9. Step 2: If a number is divisible by 9, then the sum of its digits is divisible by 9. Step 3: Since 9 = 3 × 3, any number divisible by 9 is also divisible by 3 (because if digit sum is divisible by 9, it's also divisible by 3). Step 4: Therefore, the number must be divisible by 3.

Question 7

A number is divisible by both 8 and 9. Which of the following must also divide this number?

Accepted Answer

Step 1: A number divisible by both 8 and 9 must be divisible by their LCM. Step 2: Since gcd(8,9) = 1, LCM(8,9) = 8 × 9 = 72. Step 3: Therefore, the number must be divisible by 72.

Question 8

A five-digit number 3a7b2 is divisible by 4 and 11. What is the value of a + b?

Accepted Answer

Step 1: For divisibility by 4, the last two digits (b2) must form a number divisible by 4. So b2 ∈ {12, 32, 52, 72, 92}, meaning b ∈ {1, 3, 5, 7, 9}. Step 2: For divisibility by 11, (3 + 7 + 2) − (a + b) = 12 − a − b must be divisible by 11. Step 3: 12 − a − b ≡ 0 (mod 11), so a + b ≡ 1 (mod 11). Step 4: Since a, b are single digits and b is odd, testing: if a + b = 1, then a = 0, b = 1 (b2 = 12, divisible by 4 ✓). Step 5: Verify: 30712 ÷ 4 = 7678 ✓, and (3+7+2) − (0+1) = 11 ✓.

Question 9

How many numbers between 100 and 500 are divisible by 6 but NOT divisible by 9?

Accepted Answer

Step 1: Count numbers divisible by 6 in range [100, 500]: ⌊500/6⌋ − ⌊99/6⌋ = 83 − 16 = 67. Step 2: Count numbers divisible by both 6 and 9 (i.e., divisible by lcm(6,9) = 18): ⌊500/18⌋ − ⌊99/18⌋ = 27 − 5 = 22. Step 3: Numbers divisible by 6 but NOT by 9 = 67 − 22 = 45.

Question 10

A number when divided by 12 leaves remainder 7. What remainder will it leave when divided by 4?

Accepted Answer

Step 1: If a number leaves remainder 7 when divided by 12, it can be written as n = 12k + 7 for some integer k. Step 2: Rewrite: n = 12k + 7 = 4(3k) + 7 = 4(3k + 1) + 3. Step 3: Therefore, remainder when n is divided by 4 is 3.

Question 11

A number is divisible by 15. Which of the following statements must be true?

Accepted Answer

Step 1: 15 = 3 × 5, where gcd(3,5) = 1. Step 2: If a number is divisible by 15, it must be divisible by both 3 and 5 (since they are coprime factors). Step 3: Divisibility by 3: sum of digits is divisible by 3. Divisibility by 5: last digit is 0 or 5. Step 4: Therefore, the number must be divisible by both 3 and 5, and its last digit must be 0 or 5.

SBI PO Divisibility Rules

SBI PO Divisibility Rules — Past Exam Questions

Divisibility Rules— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Divisibility Rules