If n is an odd positive integer, show that (n^2 -1 )is divisible by 8
Let n = 4q + 1 (an odd integer){tex}\\therefore \\quad n ^ { 2 } – 1 = ( 4 q + 1 ) ^ { 2 } – 1{/tex}{tex}= 16 q ^ { 2 } + 1 + 8 q – 1 \\quad \\text { Using Identity } ( a + b ) ^ { 2 } = a ^ { 2 } + 2 a b + b ^ { 2 }{/tex}{tex}= 16{q^2} + 8q{/tex}{tex}= 8 \\left( 2 q ^ { 2 } + q \\right){/tex}= 8m, which is divisible by 8.
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