Number system questions test divisibility, place value, unit digits, factors and remainders.
1Divisibility rules
2: last digit even. 3: digit sum divisible by 3. 4: last two digits divisible by 4
5: last digit 0 or 5. 8: last three digits divisible by 8. 9: digit sum divisible by 9
11: difference between sums of alternate digits is 0 or divisible by 11
2Factors
If n = pᵃ × qᵇ × rᶜ, the number of factors is (a + 1)(b + 1)(c + 1)
3Unit digits
Unit digits of powers 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; 4 and 9 alternate
4Series sums
1 + 2 + … + n = n(n + 1)/2
1² + 2² + … + n² = n(n + 1)(2n + 1)/6
1³ + … + n³ = [n(n + 1)/2]²