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The procedures for using cutting-adding method for testing a number M to be a multiple of 59 are as follows: 1 cut the units digit of M 2 add the remaining integer by r times of the deleted digit. 3 go to step 1 until the resulting integer is a multiple of 59, then M is divisible by 59. Vice versa.

First , i have no idea how to find r. then one more question is added Let M=10x+y and N=x+ry, where x, y, r, are positive integers, Prove 59/M if and only if 59/N

Please help with detailed steps!!

Tessa
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Observe that

$$6(10x+y)-(x+ry)=59x+(6-r)y\equiv (6-r)y\pmod{59}$$

$$\implies 6(10x+y) \equiv x+ry\pmod{59}$$ if $r\equiv6\pmod{59},$ the minimum positive integer value of $r$ is $6$ for all integer $x,y$

lab bhattacharjee
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  • why the modulus turn from 59 to 65? – Tessa Nov 12 '13 at 13:52
  • It should be $59$. In general, if you want to check divisibility by $p$ this way, find a multiple of $p$ that ends in $1$ or $9$. Let the tens digit of the multiple be $a$. If it is $9$, add $a+1$ times the ones digit to the rest. If it is $1$, subtract $a$ times the ones digit. So for $p=7,$ the multiple is $21, a=2$ and we subtract twice the ones digit from the rest. – Ross Millikan Nov 12 '13 at 14:17
  • @Tessa, sorry for the horrible typo – lab bhattacharjee Nov 12 '13 at 14:18