Clock arithmetic
Remainders, days of the week, and repeating cycles.
Pen and paper is fine · no calculator needed why?
Opens at level 31. You're level 1. You can read and practice here now.
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
- Replace each number with its remainder.
- Do the arithmetic on the small remainders.
- Take the remainder again. If it came out negative, add the divisor.
worked example
What is (86 × 53) mod 7? Give the remainder from 0 to 6.
- 86 leaves 2, and 53 leaves 4, when divided by 7.
- 2 × 4 = 8, and 8 leaves 1.
Answer: 1
Skip the whole weeks
- Divide the number of days by 7 and keep the remainder.
- Whole weeks land on the same weekday, so ignore them.
- Count the remainder forward from today. Today itself is day 0, not day 1.
worked 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?
- 100 = 14 × 7 + 2.
- Skip 14 weeks, then count 2 days on from Wednesday.
- That is Friday, day 5.
Answer: 5
Find the cycle
- List the remainders of the first few powers until they repeat: that is the cycle.
- Divide the exponent by the cycle length and keep the remainder.
- That remainder picks the entry in the cycle. A remainder of 0 means the last entry.
worked example
What is 2²⁰ mod 7? Give the remainder from 0 to 6.
- Powers of 2 leave 2, 4, 1, then repeat: a cycle of 3.
- 20 = 6 × 3 + 2, so take the 2nd entry.
- The answer is 4.
Answer: 4
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.
practice
Remainder arithmetic
worked example
What is (650 + 763) mod 4? Give the remainder from 0 to 3.
Answer: 1
- Reduce first: 650 leaves 2 and 763 leaves 3 when divided by 4.
- 2 + 3 = 5, and 5 = 1 × 4 + 1, so the answer is 1.
Day of the week
worked example
Today is Saturday. What day of the week will it be 371 days from now?
- Saturday
- Thursday
- Friday
- Wednesday
Answer: Saturday
- 371 = 53 × 7 + 0.
- Whole weeks bring you back to Saturday, and there are no days left over: Saturday.
Powers in cycles
worked example
What is 8⁸⁷ mod 13? Give the remainder from 0 to 12.
Answer: 5
- Powers of 8 leave these remainders when divided by 13: 8, 12, 5, 1, then the cycle repeats every 4 steps.
- 87 = 21 × 4 + 3, so 8⁸⁷ leaves the same remainder as 8³: 5.