Digits & Bases

Base representation

A number written in base $b$ with digits $d_k \cdots d_1 d_0$ represents:

To convert a base-10 number into base $b$, repeatedly divide by $b$ and record the remainders (from last to first). To convert the other way, just expand the sum above directly.

Divisibility tests

• 2: last digit is even.
• 3: digit sum divisible by 3.
• 4: last two digits form a number divisible by 4.
• 5: last digit is 0 or 5.
• 8: last three digits divisible by 8.
• 9: digit sum divisible by 9.
• 11: alternating digit sum (adding and subtracting digits in turn) divisible by 11.

The 3 and 9 tests both work because $10 \equiv 1 \pmod{3}$ and $10\equiv1\pmod9$, so every power of 10 also reduces to 1 — meaning a number is congruent to its digit sum modulo 3 or 9. The 11 test works similarly, since $10 \equiv -1 \pmod{11}$.

Practice problems