A six-digit number 11p9q4 is divisible by 24. The greatest possible value for pq is:
Show answer & explanation
Correct answer: A
Explanation
The correct answer is Option 1 i.e. 56.
Given number = 11p9q4 is divisible by 24
If a number is divisible by 24, it must be divisible by 8 and 3.
Divisibility rule of 8:- An integer is divisible by 8 only if three consecutive digits are all either 000 or constitute a number divisible by 8
According to this rule, the last three digits are divisible by 8 -
⇒ 9q4 is divisible by 8,
When we divide 9q4 by 8, for complete divisibility by 8, q should be 0, 4, or 8
We take maximum value because the question asks for maximum value
Thus, q = 8
Now,
Divisibility rule of 3:- A number is said to be divisible by 3 if the sum of all digits of that number is divisible by 3
⇒ 1 + 1 + p + 9 + q + 4 = 15 + p + q is divisible by 3.
Put q = 8,
⇒ 15 + p + 8 = 23 + p, Which is completely divisible by 3 when p = 1, 4 or 7
We take maximum values
⇒ p = 7
Hence,
⇒ pq = 7 × 8 = 56