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CBSE Class 10 Answered

For any positive integer n, prove that n^3-n is divisible by 6
Asked by dgthakur911 | 11 Jun, 2018, 08:49: PM
answered-by-expert Expert Answer
begin mathsize 16px style straight n cubed space minus space straight n space equals space straight n space left parenthesis space straight n squared space minus space 1 right parenthesis space equals space straight n space left parenthesis space straight n space minus 1 space right parenthesis space left parenthesis space straight n space plus space 1 right parenthesis If space straight a space number space is space divisible space by space 3 space comma space then space the space remainder space obtained space is space either space 0 space or space 1 space or space 2. rightwards double arrow straight n space equals space 3 straight x space or space 3 straight x space plus space 1 space or space 3 straight x space plus space 2 comma space where space straight x space is space an space Integer. If space straight n space equals space 3 straight x comma space then space straight n space is space divisible space by space 3. If space straight n space equals space 3 straight x space plus space 1 comma space then space straight n space minus 1 space equals space 3 straight x space plus 1 space minus 1 space equals space 3 straight x space is space divisible space by space 3. If space straight n space equals space 3 straight x space plus space 2 comma space then space straight n space plus space 1 space equals space 3 straight x space plus 2 space plus space 1 equals space 3 straight x space plus space 3 space equals space 3 space left parenthesis straight x space plus space 1 right parenthesis is space divisible space by space 3. space space rightwards double arrow one space of space the space numbers space among space straight n comma space straight n space minus 1 space and space space straight n space plus space 1 space is space always space divisible space by space 3. rightwards double arrow straight n space left parenthesis space straight n space minus 1 space right parenthesis space left parenthesis space straight n space plus space 1 right parenthesis space is space divisible space by space 3 space...... left parenthesis 1 right parenthesis If space straight a space number space is space divisible space by space 2 space comma space then space the space remainder space obtained space is space 0 space or space 1. rightwards double arrow straight n space equals space 2 straight y space or space 2 straight y space plus space 1 comma space where space straight y space is space an space Integer. If space straight n space equals space 2 straight y comma space then space straight n space is space divisible space by space 2. If space straight n space equals space 2 straight y space plus space 1 comma space then space straight n space minus 1 space equals space 2 straight y space plus 1 space minus 1 space equals space 2 straight y space is space divisible space by space 2. If space straight n space equals space 2 straight y space plus space 1 comma space then space straight n space plus space 1 space equals 2 straight y space plus 1 space plus 1 equals space 2 straight y space plus space 2 space is space divisible space by space 2. space space rightwards double arrow one space of space the space numbers space among space straight n comma space straight n space minus 1 space and space space straight n space plus space 1 space is space always space divisible space by space. rightwards double arrow straight n space left parenthesis space straight n space minus 1 space right parenthesis space left parenthesis space straight n space plus space 1 right parenthesis space is space divisible space by space 3 space...... left parenthesis 1 right parenthesis from space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space comma space straight n space left parenthesis space straight n space minus 1 space right parenthesis space left parenthesis space straight n space plus space 1 right parenthesis space is space divisible space by space 2 space and space space 3. Therefore comma space straight n space left parenthesis space straight n space minus 1 space right parenthesis space left parenthesis space straight n space plus space 1 right parenthesis space equals space straight n cubed space minus space straight n space is space divisible space by space 6. left parenthesis space If space straight a space number space is space divisible space by space both space 2 space ab equals nd space 3 comma space then space it space is space divisible space by space 6 ? end style
Answered by Yasmeen Khan | 12 Jun, 2018, 10:59: AM

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