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Last digits and remainders

The last digit of 17²⁰²⁶, 3¹⁰⁰ mod 7, a product mod 9: powers fall into short cycles, and the cycle gives the answer without the number.

  • Number sense
  • Untimed
  • Endless levels

How it plays

Nobody can write 17²⁰²⁶ out, but its last digit is easy: the last digits of the powers of 7 run 7, 9, 3, 1 and repeat, so only the exponent mod 4 matters. Every question here is that move — find the cycle, find the place in it.

Levels add remainders by a prime (Fermat: the exponent can be cut by multiples of p − 1), remainders by 9 from digit sums, remainders by a composite, and finally the last two digits.

Every answer is computed exactly; the working lists the cycle it was read from. The wrong options are the real slips: one place early or late in the cycle, the exponent cut by p instead of p − 1, a sum taken for a product.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes finding last digits and remainders of large powers by their cycles quicker and surer. It claims nothing wider.

Learn

How to get better at last digits and remainders: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Shrink the base first. For a last digit only the base’s last digit matters: 53 behaves like 3. For a remainder by m, replace the base by its own remainder: 22 mod 5 is 2.
  2. Write out the first few powers of that small base, by the same m, until one repeats. That run is the cycle; it is never longer than m.
  3. Divide the exponent by the cycle’s length and keep the remainder. If the cycle starts late, take off the powers before it first.
  4. Read the answer off the cycle at that place. A remainder of 0 means the last place in the cycle, not the first.

Techniques and tricks

  • The last-digit cycles

    Last digits of powers repeat in cycles of 1, 2 or 4: 2 → 2, 4, 8, 6; 3 → 3, 9, 7, 1; 7 → 7, 9, 3, 1; 8 → 8, 4, 2, 6; 4 → 4, 6; 9 → 9, 1; and 0, 1, 5, 6 never change.

    e.g. 17²⁰²⁶: 7’s cycle is 7, 9, 3, 1, and 2,026 ÷ 4 leaves 2, so the last digit is 9

  • Exponent mod 4 for any last digit

    Since every last-digit cycle has length 1, 2 or 4, cutting the exponent by fours never changes the last digit. Use 4 in place of a remainder of 0.

    e.g. 13⁵⁰: 50 ÷ 4 leaves 2, so it ends like 3² = 9

  • Fermat: cut by p − 1

    If p is prime and does not divide a, then a to the power p − 1 leaves 1. So the exponent can be cut by multiples of p − 1.

    e.g. 3¹⁰⁰ mod 7: 100 ÷ 6 leaves 4, and 3⁴ = 81 = 7 × 11 + 4, so the remainder is 4

  • Use −1 when you can

    A base one less than m is −1 by m, so its powers are just 1 and m − 1, alternating: odd powers give m − 1, even powers give 1.

    e.g. 6⁷⁷ mod 7: 6 is −1, 77 is odd, so the remainder is 6. Likewise 99³⁷ ends in 99 and 99³⁸ ends in 01.

  • Digit sums for 9 and 3

    10 leaves 1 on division by 9, so a number leaves what its digit sum leaves. Remainders then add with the sum and multiply with the product; reduce again at the end.

    e.g. 523 × 718 mod 9: 5 + 2 + 3 = 10 → 1, and 7 + 1 + 8 = 16 → 7. 1 × 7 = 7.

  • Tens digit for a base ending in 1

    For a base 10k + 1, the last two digits of its n-th power are the last digit of k × n, then 1.

    e.g. 21³⁷: 2 × 37 = 74, so it ends in 41

  • Cycles mod 100 from small ones

    Some bases cycle quickly by 100. 7⁴ = 2,401 ends in 01, so 7’s last two digits run 07, 49, 43, 01 and repeat every four.

    e.g. 7⁸³: 83 ÷ 4 leaves 3, so it ends in 43

As the levels climb

  • First, last digits of single powers. Learn the short cycles above until you can recite them; the rest is dividing the exponent by 4.
  • Then remainders by a prime (3, 5, 7, 11 or 13) and remainders of sums and products by 9 or 3 arrive together. For the primes, cut the exponent by p − 1; for the nines, add digits.
  • Then remainders by a composite: 4, 6, 8, 9, 12, 15, 16 or 20. Fermat does not apply, so walk the powers. Watch for cycles that start late — the powers of 2 by 8 run 2, 4, then 0 for ever.
  • Last, the last two digits: a cycle by 100. Look for a base ending in 1, a base of 99 or 49, or a small power that ends in 01, before you start walking.
  • Between the new kinds, the bases and exponents keep growing — exponents into the thousands, products of six-digit numbers. The method does not change; only the division of the exponent gets longer. You get a little more time each step.

Traps to avoid

  • One place out in the cycle. The places just before and after the right one are the first wrong options the game builds. Count from place 1, and use the last place when the division leaves 0.
  • Cutting the exponent by p instead of p − 1. For 5¹⁰⁰ mod 7, cutting by 7 leaves 5² = 25, remainder 4. Cutting by 6 leaves 5⁴ = 625, remainder 2 — the true answer.
  • Using the exponent’s last digit, or the base’s, as the answer. Both turn up among the options: the answer is in the cycle, not in the numbers on the page.
  • A sum taken for a product. For a remainder of a product, multiply the two remainders; for a sum, add them. The other one is usually among the options.
  • For the last two digits, answering with the last digit only, or cutting the exponent by 10. Neither follows the cycle by 100.

How to practise

  • Use “Kinds of question” to drill “Last digit of aⁿ” on its own until the cycles are instant, then add “Powers mod a prime”.
  • Switch “Answer by” to “Typing the answer” once choosing is easy: with no neighbouring places on show, you must count the cycle exactly.
  • Keep “Options shown” at 4 while learning, then raise it to 8: with most of the digits or remainders on offer, only the method can pick the right one.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More number sense

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