Divisibility Quiz

Which of the following numbers is divisible completely by both 9 and 11?

• 277218

• 181998

• 10098

• 12345

Answer explanation

To check divisibility by both 9 and 11, we find the least common multiple (LCM), which is 99. Among the options, 10098 is divisible by 99 (10098 ÷ 99 = 102), while the others are not. Thus, 10098 is the correct answer.

Which of the following numbers is NOT divisible by 9?

• 77832

• 45390

• 35253

• 49104

Answer explanation

To check divisibility by 9, sum the digits of each number. For 45390, 4+5+3+9+0 = 21, which is not divisible by 9. The other numbers' digit sums are divisible by 9, making 45390 the correct answer.

Which of the following numbers is NOT divisible by 8?

• 35792

• 35112

• 35412

• 35552

Answer explanation

To check divisibility by 8, the last three digits of a number must be divisible by 8. For 35412, the last three digits are 412, and 412 ÷ 8 = 51.5, which is not an integer. Thus, 35412 is NOT divisible by 8.

If the 7 digit number 504x5y3 is divisible by 11, then one of the values of the sum of x and y is:

• 11

• 5

• 17

• 7

Answer explanation

For a number to be divisible by 11, the difference between the sum of its digits in odd positions and the sum of its digits in even positions must be a multiple of 11. Here, x + y = 17 satisfies this condition.

If 11-digit number 88p554085k6, k ≠ p, is divisible by 72, then what is the value of (3k + 2p)?

• 12

• 7

• 13

• 23

Answer explanation

For divisibility by 72, the number must be divisible by both 8 and 9. The last three digits (k6) must form a number divisible by 8, and the sum of the digits must be divisible by 9. Solving gives k=3 and p=2, leading to 3k + 2p = 13.

Find the remainder, when 171 × 172 × 173 is divided by 17.

• 6

• 8

• 9

• 7

Answer explanation

To find the remainder of 171 × 172 × 173 when divided by 17, we can reduce each number modulo 17: 171 ≡ 1, 172 ≡ 2, 173 ≡ 3. Thus, (1 × 2 × 3) mod 17 = 6. Therefore, the remainder is 6.

When a number is divided by a divisor, the remainder is 16. When twice the original number is divided by the same divisor, the remainder is 3. Find the value of that divisor.

• 23

• 29

• 53

• 51

Answer explanation

Let the number be N. From the problem, we have N % d = 16 and (2N) % d = 3. This implies 2N = k*d + 3. Substituting N = m*d + 16 gives 2(m*d + 16) % d = 3. Solving leads to d = 29.

If the number 6484y6 is divisible by 8, then find the least value of y?

• 7

• 1

• 4

• 3

Answer explanation

For a number to be divisible by 8, the last three digits must form a number divisible by 8. Here, the last three digits are 'y6'. Testing values, '16' (when y=1) is divisible by 8, making 1 the least value for y.

If the 15 digit number 4a5124356789734 is divisible by 9, then the value of 'a' is:

• 5

• 1

• 4

• 3

Answer explanation

A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits in 4a5124356789734 is 46 + a. For 46 + a to be divisible by 9, a must be 4, making the total 50, which is divisible by 9.

If the 8 digit number 3x5479y4 is divisible by 88 and the 8 digit number 425139z2 is divisible by 9, then find the maximum possible value of (3x + 2y – z).

• 37

• 25

• 35

• 33

Answer explanation

For 3x5479y4 to be divisible by 88, y must be 0 or 8, and x must be 0, 2, 4, 6, or 8. For 425139z2 to be divisible by 9, z must be 0, 3, 6, or 9. The maximum value of (3x + 2y - z) is 33 when x=8, y=8, z=9.