Question

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Q.Use Euclid's Division Lemma to show that the square of any positive integer cannot be of the form 5m+2 or 5m+3 for some integer m.



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Answer 1

Answer:

The square of any positive integer can be of the form 5m, 5m + 1, 5m + 4 for some integer m but it cannot be of the form 5m + 2 or 5m + 3 where m is any positive integer.

Step-by-step explanation:

According to Euclid's division lemma, any positive integer can be of the form of a = bq + r where, 0 ≤ r < b.

In this case, b = 5 then, r = 0, 1, 2, 3, 4

Thus, any positive integer can be of the form of 5m, 5m + 1, 5m + 2, 5m + 3, 5m + 4.

Now, ${(5q)}^{2} = {(25q)}^{2} = 5{(5)}^{2} = 5m$, where m=${(5q)}^{2}$, which is an integer.

$(5q+1)^{2} = 25q^{2}+10q+1 = 5(5q^{2} +2q)+1= 5m+1$, where m = $5q^{2} +2q$, which is an integer.

$(5q+2)^{2} = 25q^{2}+20q+4 = 5(5q^{2} +4q)+4= 5m+4$, where m = $5q^{2} +4q$, which is an integer.

$(5q+3)^{2} = 25q^{2}+30q+9 = 5(5q^{2} +6q+1)+4= 5m+4$, where m = $5q^{2} +6q+1$, which is an integer.

$(5q+4)^{2} = 25q^{2}+40q+16 = 5(5q^{2} +8q+3)+1= 5m+1$, where m = $5q^{2} +8q+3$, which is an integer.

Thus, the square of any positive integer is of the form 5m, 5m + 1, 5m + 4 for some integer m.

It follows that the square of any positive integer cannot be of the form 5m + 2 or 5m + 3 for some integer m.