is every odd integer is in form 4k+1 ?
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By definition, a = 2q is even. Let q = 2k. Then a = 4k. So, a + 1 = 4k + 1 and a + 3 = 4k + 3.
a + 5 = 4k + 4 + 1 = 4(k + 1) + 1
a + 7 = 4k + 4 + 3 = 4(k + 1) + 3
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yes
let n be any arbitory odd positive integer.
On dividing n by 4, let m be the quotient and r be the remainder.
So, by Euclid's division lemma, we have :
n=4m+r, where 0<_r<4.
n=4m or (4m+1).
Clearly, 4m is even and since n is odd, so n cannot be 4m.
therefore, n=(4m+1), for some integer m.
let n be any arbitory odd positive integer.
On dividing n by 4, let m be the quotient and r be the remainder.
So, by Euclid's division lemma, we have :
n=4m+r, where 0<_r<4.
n=4m or (4m+1).
Clearly, 4m is even and since n is odd, so n cannot be 4m.
therefore, n=(4m+1), for some integer m.
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