   Chapter 2.5, Problem 34E

Chapter
Section
Textbook Problem

If m is an integer, show that m 2 is congruent modulo 8 to one of the integers 0 , 1 , or 4 . (Hint: Use the Division Algorithm, and consider the possible remainders in m = 4 q +   r .)

To determine

To prove: If m is an integer, then m2 is congruent modulo 8 to one of the integers 0,1, or 4.

Explanation

Given information:

m is an integer.

Formula used:

1) Definition: Congruence Modulo n

Let n be a positive integer, n>1. For integers x and y, x is congruent to y modulo n, if and only if xy is a multiple of n. We write xy(modn) to indicate that x is congruent to y modulo n.

2) Theorem: The Division Algorithm:

Let a and b be integers with b>0. Then, there exist unique integers q and r, such that a=bq+r with 0r<b.

Proof:

If m is an integer, then there are two cases as follows:

Case 1: m is an even integer:

If m is an even integer, then there exists k such that m=2k.

Therefore, m2=(2k)2=4k2 …… (1)

Subcase i: k is an even integer:

If k is an even integer, then clearly k2 is even.

Therefore, k2=2k1 for k1.

Substitute this in equation (1),

m2=4(2k1)

m20=8k1

Since m20 is a multiple of 8, then by using the definition,

m20(mod8)

Subcase ii: k is an odd integer:

If k is an odd integer, then clearly k2 is odd.

Therefore, k2=2k1+1 for k1

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