Does this Pattern Continue?

Number Theory Level 3

The divisibility rule of 2 is that the last digit of a number is divisible by 2.

The divisibility rule of 4 is that the last 2 digits of a number are divisible by 4.

The divisibility rule of 8 is that the last 3 digits of a number are divisible by 8.

Is it true that for any number x that is a power of 2, the divisibility rule of x is that the last $\log_2 x$ digits of any number are divisible by x?

No Yes

1 solution

Jordan Cahn
Dec 21, 2018

Let $x=2^n$ . Let $a$ be a number for which we want to test its divisibility by $x$ . The "last $n$ digits of $a$ " can be expressed as $a \bmod 10^n$ . However, $10^n = 2^n \cdot 5^n$ . So $a \equiv 0\bmod 10 \iff a\equiv 0\bmod 2^n$

Does this Pattern Continue?

1 solution

0 pending reports