Divisibility checks you can do on sight
A number-system question often asks only whether a large number divides evenly. You do not need the quotient. These checks are enough for the usual tests:
- 2: last digit is even.
- 3 or 9: the sum of the digits is divisible by 3 or by 9.
- 4: the last two digits form a number divisible by 4.
- 5: the last digit is 0 or 5.
- 8: the last three digits form a number divisible by 8.
- 11: the alternating sum of the digits is a multiple of 11, including 0.
For 918,291 and divisibility by 11: 9 − 1 + 8 − 2 + 9 − 1 = 22. Since 22 is a multiple of 11, the whole number is divisible by 11. For 5,736 and divisibility by 8, look only at 736. Half of 736 is 368, half again is 184, half again is 92, so 736 is divisible by 8 and so is 5,736.
Unit digits repeat in a short cycle
The units digit of a power depends only on the units digit of the base, and that digit repeats. You never need the full power.
- Powers of 2: 2, 4, 8, 6, then repeat. Cycle length 4.
- Powers of 3: 3, 9, 7, 1. Cycle length 4.
- Powers of 7: 7, 9, 3, 1. Cycle length 4.
- Powers of 4: 4, 6. Cycle length 2. Powers of 9: 9, 1. Cycle length 2.
- Powers of 5 end in 5. Powers of 6 end in 6.
Divide the exponent by the cycle length and use the remainder as the position in the cycle. If the remainder is 0, use the last digit of the cycle.
Units digit of 723: 23 ÷ 4 leaves remainder 3, so take the third entry, which is 3. Units digit of 210: 10 ÷ 4 leaves remainder 2, so take the second entry, which is 4. That matches 1,024. Units digit of 320: 20 ÷ 4 leaves remainder 0, so take the last entry of the cycle, which is 1.
Remainders use the same cycle
Finding 210 divided by 5 is the same habit with a different cycle. Powers of 2 modulo 5 run 2, 4, 3, 1 and then repeat. The tenth power sits at position 2 in that cycle, so the remainder is 4. You can check: 1,024 = 5 × 204 + 4.
When the divisor is small, write the first few remainders until the list repeats, then map the exponent onto that list. Do not expand the power.
What to drill
Memorize the divisibility list and the unit-digit cycles for 2, 3, 4, 7, 8, and 9. In a timed paper, the win is noticing that the question is a cycle problem and stopping after the remainder of the exponent. If the remainder of the exponent is 0, take the end of the cycle, not the first entry.