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When a positive integer m^2 is divided by 4, what is the remainder?
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27 Mar 2017, 05:55

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MathRevolution wrote:

When a positive integer m² is divided by 4, what is the remainder?

1) When m is divided by 3, the remainder is 1 2) When m is divided by 2, the remainder is 1

Target question:What is the remainder when m² is divided by 4?

Statement 1: When m is divided by 3, the remainder is 1 Let's TEST some values. There are several values of m that satisfy statement 1. Here are two: Case a: m = 4, which means m² =16. In this case, the remainder is 0 when m² is divided by 4 Case b: m = 7, which means m² =49. In this case, the remainder is 1 when m² is divided by 4 Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: When m is divided by 2, the remainder is 1 ASIDE: There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R" For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2 Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

So, in this case, we can write: m = 2k + 1 (where k is some integer) If m = 2k + 1, then m² = (2k + 1)² = 4k² + 4k + 1 = 4(k² + k) + 1 In other words, m² = 4(some integer) + 1 Since m² is 1 GREATER THAN some multiple of 4, we can conclude that the remainder is 1 when m² is divided by 4 Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Re: When a positive integer m^2 is divided by 4, what is the remainder?
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29 Mar 2017, 00:43

==> In the original condition, there is 1 variable (m) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer. For remainder questions, it is best to use direct substitution.

For con 1), if you substitute from m=3p+1=1,4,7…, from m=1, 1^2=1=4(0)+1, the remainder=1, and if m=4, from 4^2=16=4(4)+0, the remainder=0, hence it is not unique and not sufficient. For con 2), from m=2q+1=1,3,5,7,…, you get m2=1,9,25,49.., and the remainder divided by 4 always becomes 1, hence it is unique and sufficient.

Therefore, the answer is B. Answer: B
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Re: When a positive integer m^2 is divided by 4, what is the remainder?
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05 Jul 2018, 06:36

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