Divisibility without calculations

Divisibility tests learned at school -------------------------------

Let us quickly review the most familiar tests for determining, with little or no calculation and without a calculator, whether an integer N = 10d + u, where u is the units digit and d the number of tens in N, is divisible by some simple numbers. Congruences naturally take center stage. We use the notation N $\equiv \backslash equiv$ n [p], read as "N is congruent to n modulo p," meaning that N and n have the same remainder when divided by p.

For 2*ⁿ or 5ⁿ: look at the last n* digits --------------------------------------------------

We all remember the divisibility tests for 2 and 5: the last digit is decisive! If it is even, then the integer itself is even; if it is 0 or 5, then the integer is divisible by 5. Since N = 10d + u, we have N $\equiv \backslash equiv$ u [2] and N $\equiv \backslash equiv$ u [5], so N is divisible by 2 or 5 if and only if its units digit is.