Solved Examples

Example 1: Check whether 46,332 is divisible by 4, 6, and 9. - Last two digits = 32 → 32 ÷ 4 = 8, exact → divisible by 4 - Digit sum = 4+6+3+3+2 = 18 → divisible by 3 and 9. Since divisible by both 2 (even) and 3 → divisible by 6. Answer: Divisible by 4, 6, and 9 (also by 3, 2, 18)

Example 2: Is 9,834 divisible by 7? - Last digit = 4, double it = 8. Remaining number = 983. 983 − 8 = 975. - Repeat: last digit of 975 = 5, double = 10. Remaining = 97. 97 − 10 = 87. - 87 is not divisible by 7 (7×12=84, remainder 3). Answer: 9,834 is NOT divisible by 7

Example 3: Is 5,16,846 divisible by 13? - Last digit = 6, ×4 = 24. Remaining number = 51684. 51684 + 24 = 51708. - Repeat: last digit 8, ×4=32. Remaining = 5170. 5170+32 = 5202. - Repeat: last digit 2, ×4=8. Remaining=520. 520+8=528. - Repeat: last digit 8,×4=32. Remaining=52. 52+32=84. - 84 ÷ 13 = 6.46 — not exact, so 84 is not divisible by 13. Answer: 5,16,846 is NOT divisible by 13 (Note: for exam speed, if the number after 2–3 iterations is small and not obviously a multiple, it is usually faster to just check by estimation/direct division for numbers under 200.)

Example 4: Check whether 918,082 is divisible by 11. - Digits from right: 2(pos1), 8(pos2), 0(pos3), 8(pos4), 1(pos5), 9(pos6) - Sum at odd positions (1,3,5) = 2+0+1 = 3 - Sum at even positions (2,4,6) = 8+8+9 = 25 - Difference = 25 − 3 = 22, which is a multiple of 11. Answer: Divisible by 11

Example 5 (Finding a missing digit): Find the value of digit x if the number 3x572 is divisible by 9. - Digit sum = 3 + x + 5 + 7 + 2 = 17 + x - For divisibility by 9, 17 + x must be a multiple of 9. The nearest multiples of 9 near 17 are 18 and 27. - 17 + x = 18 → x = 1 (valid, since x must be a single digit 0–9) - 17 + x = 27 → x = 10 (invalid, not a single digit) Answer: x = 1 (the number is 31,572)

Example 6 (Missing digit with 11): Find the digit y if 4y3812 is divisible by 11 (y is the digit in the lakhs-adjacent place as written). - Number: 4, y, 3, 8, 1, 2 (left to right). From the right, positions are: 2(1), 1(2), 8(3), 3(4), y(5), 4(6). - Odd positions (1,3,5): 2 + 8 + y = 10 + y - Even positions (2,4,6): 1 + 3 + 4 = 8 - Difference = (10 + y) − 8 = 2 + y, which must be 0 or a multiple of 11. - Since y is a digit (0–9), 2 + y = 11 → y = 9 Answer: y = 9 (the number is 493812)

Example 7 (Missing digit with 8): Find the digit x if the number 34526x is divisible by 8. - Divisibility by 8 depends only on the number formed by the last three digits: “26x” (i.e., 260 + x). - 260 ÷ 8 = 32 remainder 4, so 260 ≡ 4 (mod 8). We need 260 + x ≡ 0 (mod 8), i.e., x ≡ 4 (mod 8). - Since x is a single digit (0–9), x = 4 (x = 12 is not a valid digit). Answer: x = 4 (last three digits become 264, and 264 ÷ 8 = 33 exactly)

Example 8 (Combined divisibility — by 4 and 9 together): Find the digit z such that the number 2z396 is divisible by both 4 and 9 (i.e., by 36). - Last two digits are “96” regardless of z, and 96 ÷ 4 = 24 exactly → the number is divisible by 4 no matter what z is. - For divisibility by 9: digit sum = 2 + z + 3 + 9 + 6 = 20 + z, which must be a multiple of 9. The nearest multiple of 9 above 20 is 27. - 20 + z = 27 → z = 7 Answer: z = 7 (the number is 27,396; check: 27396 ÷ 4 = 6849 exact, and digit sum 2+7+3+9+6 = 27, divisible by 9 → the number is divisible by 36)