Shave off Last Two
Number Theory
Level
3
One can test if an integer \(n\) is divisible by prime \(101\) by subtracting the last two digits (as a number) from \(n\) with those digits shaved off, and see if the result is divisible by \(101\). For example, \(162794931\) is divisible by \(101\) because \(1627949 - 31 = 1627918\) and \(101 \, | \, 1627918\).
What are the smallest factors you should multiply the last two digits of \(n\) (as a number) with, before subtracting from the shaved-off \(n\) in the divisibility tests for \(p = 43\) and \(p = 67?\)
Give your answer as the product of the factors.
by
Borut Levart