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Question
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
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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Solution
Number divisible by 5 can be of the form:-
d = 5k + r,
where 0 ≤ r <5
If d = 5k, d² = 5.m,
where m is some integer and m = 5k²
If d = 5k + 1 , d² = 5m + 1
If d = 5k + 2 , d² = 5m + 4
If d = 5k + 3 , d² = 5m + 4
If d = 5k + 4, d² = 5m + 1
Therefore, the square of any positive integer cannot be in the form of 5m + 2 or 5m + 3 for any integer "m".
d = 5k + r,
where 0 ≤ r <5
If d = 5k, d² = 5.m,
where m is some integer and m = 5k²
If d = 5k + 1 , d² = 5m + 1
If d = 5k + 2 , d² = 5m + 4
If d = 5k + 3 , d² = 5m + 4
If d = 5k + 4, d² = 5m + 1
Therefore, the square of any positive integer cannot be in the form of 5m + 2 or 5m + 3 for any integer "m".
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