Question

If the number 6523678pq is divisible by 99, the missing digits p and q are :

• A. p = 4, q = 4
• B. p = 6, q = 2
• C. p = 2, q = 6
• D. p = 3, q = 5

Answer 1

Answer:

C. p = 2, q = 6

Solution: Given: The number 6523678pq is divisible by 99. Concept used: Divisibility rule of 9 ⇒ for any number to be divisible by 9, the sum of its digit should be divisible by 9. Divisibility rule of 11 ⇒ the subtraction of alternate digits of the number should add up to zero or be divisible by 11. Calculation: According to the question, 6523678pq is divisible by 99 This means it is divisible by (11 × 9) ∴ It is divisible by both 11 and 9 Divisibility by 9, ⇒ (6 + 5 + 2 + 3 + 6 + 7 + 8 + p + q) should be divisible by 9 ⇒ (37 + p + q) is divisible by 9 ⇒ p + q = 8 .....(1) [ The nearest multiple of 9 greater than 37 is 45. Hence we have to add 8 to the 37 to make it a multiple of 9] Divisibility by 11, ⇒ (6 + 2 + 6 + 8 + q) - (5 + 3 + 7 + p) = 11 ⇒ 22 + q - 15 - p = 11 ⇒ q - p = 11 - 7 = 4 .....(2) Adding (1) and (2), we get ⇒ 2 × q = 12 ⇒ q = 6 p = 8 - q = 8 - 6 ⇒ p = 2 ∴ The value of p = 2 and q = 6.