If for some positive integer k, 543 + 5k is divisible by 7, then which one of the following could be the value of k ?

To determine the value of k, we use modular arithmetic. We are given that (543 + 5k) is divisible by 7, which means:

(543 + 5k) ≡ 0 (mod 7)

First, find the remainder when 543 is divided by 7:
543 = (7 × 77) + 4
So, 543 ≡ 4 (mod 7).

Substituting this into the congruence:
4 + 5k ≡ 0 (mod 7)
5k ≡ -4 (mod 7)
To make the remainder positive, add 7: 5k ≡ 3 (mod 7).

Now, we test the given options for k:
A) k = 42: 5 × 42 = 210. 210 ÷ 7 = 30 (Remainder 0).
B) k = 44: 5 × 44 = 220. 220 ÷ 7 = 31 with a remainder of 3. (Correct)
C) k = 46: 5 × 46 = 230. 230 ÷ 7 = 32 (Remainder 6).
D) k = 48: 5 × 48 = 240. 240 ÷ 7 = 34 (Remainder 2).

Since k = 44 satisfies the condition 5k ≡ 3 (mod 7), it is the correct value.