Most number-system questions use a handful of facts.
Divisibility rules
| By | Rule |
|---|---|
| 3 / 9 | Sum of digits divisible by 3 / 9 |
| 4 / 8 | Last 2 / last 3 digits divisible by 4 / 8 |
| 11 | (Sum of odd-place digits) − (sum of even-place digits) is 0 or a multiple of 11 |
HCF and LCM. For two numbers, HCF × LCM = product of the numbers. This does not hold for three numbers.
Number of factors. Write N = a^p × b^q × c^r (prime powers). N has (p+1)(q+1)(r+1) factors.
Unit digit of a power. Unit digits repeat in cycles of at most 4: 2 → 2, 4, 8, 6; 3 → 3, 9, 7, 1; 7 → 7, 9, 3, 1; 8 → 8, 4, 2, 6. 0, 1, 5 and 6 always end in themselves. Divide the power by 4 and use the remainder (remainder 0 means the 4th term).
Worked example. Unit digit of 7^95:
cycle of 7: 7, 9, 3, 1
95 ÷ 4 leaves remainder 3 -> third term = 3The answer is 3.
Trap: when the remainder is 0, students pick the first term. Remainder 0 means the last term of the cycle (for 7, that is 1).