InterviewSolution
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InterviewSolution
| 1. |
Use Euclid\'s division leema to show square of any positive integer are either in form 4q ,4q+1 |
| Answer» Let a = 4q + r, when r = 0, 1, 2 and 3{tex}\\therefore{/tex}Numbers are 4q, 4q + 1, 4q + 2 and 4q + 3{tex}( a ) ^ { 2 } = ( 4 q ) ^ { 2 } = 16 q ^ { 2 } = 4 ( 4 q ) ^ { 2 } = 4 m{/tex}{tex}( a ) ^ { 2 } = ( 4 q + 1 ) ^ { 2 } = 16 q ^ { 2 } + 8 q + 1 = 4 \\left( 4 q ^ { 2 } + 2 q \\right) + 1 = 4 m + 1{/tex}{tex}( a ) ^ { 2 } = ( 4 q + 2 ) ^ { 2 } = 16 q ^ { 2 } + 16 q + 4 = 4 \\left( 4 q ^ { 2 } + 4 q + 1 \\right) = 4 m{/tex}{tex}( a ) ^ { 2 } = ( 4 q + 3 ) ^ { 2 } = 16 q ^ { 2 } + 24 q + 9 = 4 \\left( 4 q ^ { 2 } + 6 q + 2 \\right) + 1 = 4 m + 1{/tex}{tex}\\therefore {/tex}\xa0the square of any +ve integer is of the form 4q or 4q + 1\xa0 | |