Question:
An integer x when divided by 5 leaves the remainder 3. Prove that when 4x is divided by 5, 2 is the remainder

Hi,

If x, when divided by 5, leaves a remainder of 3 then x is five times an integer, plus 3. That is

x = (5 k) + 3

for some integer k. Then 4x is

4 x = (4 5 k) + (4 3) = (20 k) + 12 = (20 k) + 10 + 2

Thus

 ^{(4 x)}/₅ = ²⁰/₅ k + ¹⁰/₅ + ²/₅ = 4 k + 2 + ²/₅

Thus the remainder, when 4x is divided by 5, is 2.

Penny

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