For how many different values of $k$ is the 4 -digit number $7 k 52$ divisible by 12 ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
For how many different values of $k$ is the 4 -digit number $7 k 52$ divisible by 12 ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

## Solution

Since $12=4 \times 3$ the number $7 k 52$ must be divisible by both 4 and 3 . Since 52 is the number formed by the last two digits divisible by 4 then we need only ask, 'for what values of $k$ is $7 k 52$ divisible by 3?' If a number is divisible by 3 the sum of its digits must be a multiple of 3 . Thus $7+k+5+2$ or $14+k$ must be a multiple of 3 . The only acceptable values for $k$ are 1,4 or 7 .

Thus, are three values.

ANSWER: (D)