doing math

courses › Number Theory

Clock arithmetic

level 31 course

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)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

Clock arithmetic keeps only remainders. Five hours after 9 o'clock is 2 o'clock, because 14 hours leaves 2 once you take out 12. We write 14 mod 12 for that remainder.

The key fact: you can take remainders first, then add, subtract or multiply, then take the remainder again. The answer is the same, and the numbers stay small.

Techniques

Reduce first

Adding, subtracting or multiplying big numbers, then taking a remainder.

  1. Replace each number with its remainder.
  2. Do the arithmetic on the small remainders.
  3. Take the remainder again. If it came out negative, add the divisor.
worked example

Example: What is (86 × 53) mod 7? Give the remainder from 0 to 6.

  1. 86 leaves 2, and 53 leaves 4, when divided by 7.
  2. 2 × 4 = 8, and 8 leaves 1.

Answer: 1

Skip the whole weeks

The day of the week some number of days from today.

  1. Divide the number of days by 7 and keep the remainder.
  2. Whole weeks land on the same weekday, so ignore them.
  3. Count the remainder forward from today. Today itself is day 0, not day 1.
worked example

Example: Number the days of the week Sunday 0 to Saturday 6. Today is Wednesday, day 3. Which day number is 100 days from now?

  1. 100 = 14 × 7 + 2.
  2. Skip 14 weeks, then count 2 days on from Wednesday.
  3. That is Friday, day 5.

Answer: 5

Find the cycle

A big power, like 2¹⁰⁰, divided by a small number.

  1. List the remainders of the first few powers until they repeat: that is the cycle.
  2. Divide the exponent by the cycle length and keep the remainder.
  3. That remainder picks the entry in the cycle. A remainder of 0 means the last entry.
worked example

Example: What is 2²⁰ mod 7? Give the remainder from 0 to 6.

  1. Powers of 2 leave 2, 4, 1, then repeat: a cycle of 3.
  2. 20 = 6 × 3 + 2, so take the 2nd entry.
  3. The answer is 4.

Answer: 4

Tips by skill

  • TipRemainder arithmetic: Replace each number with its remainder, do the arithmetic, then take the remainder again. Add the divisor if it goes negative.
  • TipDay of the week: Divide the days by 7. Skip the whole weeks and count the remainder forward from today.
  • TipPowers in cycles: List the remainders of the powers until they cycle. The exponent's remainder by the cycle length picks the entry; 0 means the last.

Watch out for

  • Reducing the numbers but not the result. If the remainders multiply to 6 and you are dividing by 5, the answer is 1, not 6.
  • Counting today as day 1. From Tuesday, 2 days on is Thursday; counting Tuesday itself lands a day short.
  • Landing one step off in the cycle. A remainder of 1 means the first entry, and 0 means the last.

skills · practice stats

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

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.