courses › Number Theory

Remainders, days of the week, and repeating cycles.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 31.

Sign in to start

Builds on: Divisibility (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Remainder arithmetic par 14 s not tried yet

  worked example

  What is (650 + 763) mod 4? Give the remainder from 0 to 3.

  Answer: 1

  1. Reduce first: 650 leaves 2 and 763 leaves 3 when divided by 4.
  2. 2 + 3 = 5, and 5 = 1 × 4 + 1, so the answer is 1.

• Day of the week par 17 s not tried yet

  worked example

  Today is Saturday. What day of the week will it be 371 days from now?

  1. Saturday
  2. Thursday
  3. Friday
  4. Wednesday

  Answer: Saturday

  1. 371 = 53 × 7 + 0.
  2. Whole weeks bring you back to Saturday, and there are no days left over: Saturday.

• Powers in cycles par 25 s not tried yet

  worked example

  What is 8⁸⁷ mod 13? Give the remainder from 0 to 12.

  Answer: 5

  1. Powers of 8 leave these remainders when divided by 13: 8, 12, 5, 1, then the cycle repeats every 4 steps.
  2. 87 = 21 × 4 + 3, so 8⁸⁷ leaves the same remainder as 8³: 5.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

your rounds

No rounds yet.