An integer m ≥ 0 satisfies the condition that 5m leaves a remainder of 30 when divided by 84. Which of the following represents all possible remainders when 8 m is divided by 84?

We are given that 5m leaves a remainder of 30 when divided by 84, so we can write 5m equals 84k plus 30 for some integer k. Factoring out a 6 gives 5m equals 6 times (14k plus 5), which means m must be a multiple of 6, so let m equal 6y. Substituting this into the equation gives 30y equals 14k plus 5, but since the left side is even and the right side is odd, k must be odd. Letting k equal 1 gives y equaling 19 over 30 which is not an integer, so instead we multiply to find the direct relation by noticing 8m is (8/5) of 5m, meaning 8m equals 8/5 times (84k plus 30). The values of 84k plus 30 must be multiples of 5, meaning 84k must end in 0, requiring k to be a multiple of 5. Let k equal 5, then 5m equals 450, meaning m equals 90, and 8m equals 720. Dividing 720 by 84 gives 8 with a remainder of 48, so the only possible remainder is 48.

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