courses › Number Theory

Quick tests for divisibility, and counting divisors.

Best in your head · no pen, no calculator why?

Learn first (about 3 minutes)

Opens at level 16.

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Builds on: Factors & primes (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 by 9 par 11 s not tried yet

  worked example

  What is the remainder when 90,117 is divided by 3?

  Answer: 0

  1. Digit sum: 9 + 0 + 1 + 1 + 7 = 18.
  2. A number leaves the same remainder by 3 as its digit sum: 18 = 3 × 6 + 0, so the remainder is 0.

• Divisibility tests par 12 s not tried yet

  worked example

  Which number is divisible by 6?

  1. 9,400
  2. 2,088
  3. 1,821
  4. 4,502

  Answer: 2,088

  1. Divisible by 6 means even and divisible by 3. 2,088 is even, and its digit sum 2 + 0 + 8 + 8 = 18 is a multiple of 3.
  2. Each of the others fails one of the two tests.
  3. So the answer is 2,088.

• Count the divisors par 20 s not tried yet

  worked example

  How many positive divisors does 63 have?

  Answer: 6

  1. 63 = 3² × 7.
  2. A divisor uses one power of each prime: 3⁰ to 3² (3 choices) and 7⁰ to 7¹ (2 choices).
  3. 3 × 2 = 6 divisors.

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

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