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| 1. |
Express the hcf of 963 and 657 as 963x and 65 7y where x and y are integers |
| Answer» By applying Euclid’s division lemma,{tex}963 = 657\\times 1 + 306.{/tex}Since remainder ≠ 0, apply division lemma on divisor 657 and remainder 306{tex}657 = 306\\times 2 + 45.{/tex}Since remainder ≠ 0, apply division lemma on divisor 306 and remainder 45{tex}306 = 45\\times 6 + 36.{/tex}Since remainder ≠ 0, apply division lemma on divisor 45 and remainder 36{tex}45 = 36\\times 1 + 9.{/tex}Since remainder ≠ 0, apply division lemma on divisor 36 and remainder 9{tex}36 = 9 \\times\xa04 + 0.{/tex}Therefore, {tex}H.C.F. = 9.{/tex}HCF of 2 numbers can be expressed as the linear combination of the numbers{tex}\\Rightarrow {/tex} 9 = 45 - 36 {tex}\\times{/tex}\xa01= {tex}45 - [306 - 45 \\times\xa06] \\times\xa01 = 45 - 306\\times 1 + 45\\times 6{/tex}= {tex}45 \\times\xa07 - 306\\times 1 = [657 -306\\times 2]\\times 7 - 306 × 1{/tex}= {tex}657 \\times\xa07 - 306 \\times\xa014 - 306 × 1{/tex}= {tex}657\\times 7 - 306\\times 15{/tex}= {tex}657\\times 7 - [963 - 657 × 1] \\times\xa015{/tex}= 657 {tex}\\times{/tex}\xa07 - 963 {tex}\\times{/tex}\xa015 + 657 {tex}\\times{/tex}\xa015= 657 {tex}\\times{/tex}\xa022 - 963 {tex}\\times{/tex}\xa015.Hence, obtained. | |