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In a division, if the (the number which is being divided) is multiplied by certain factor and then divided by the same divisor, then the new remainder will be obtained by multiplying the original remainder by the same factor with which the dividend has been multiplied.

For example, when 11 is divide by 8, the remainder is 3. When the dividend 11 is multiplied by 2, we get 22 and when this number is divided by 8, the remainder is 6 which is same as the original remainder 3 multiplied by 2.

Is it true ???

Because I tried 111/7, the remainder is 6. Then I multiplied 111 by 2 (222) and then divided it by 7, but the remainder was 5. Could any explain me this??

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  • $\begingroup$ $222=2\cdot111, 111\equiv6\pmod 7\implies 222\equiv 6\cdot2\pmod 7\equiv5$ $\endgroup$ – lab bhattacharjee Jul 1 '13 at 15:06
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Hint: $6\cdot2=12$, and when $12$ is divided by $7$, the remainder is $5$. What you said is true in general if you take the remainder once again, to ensure that the final answer is less than what you divide by.

See Modular Arithmetic for more information.

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