courses › Number Theory

Divisibility

level 16 course

Quick tests for divisibility, and counting divisors.

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

Opens at level 16.

Builds on: Factors & primes (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

The digits of a number can tell you what a division leaves, without doing the division.

Every place value (10, 100, 1,000 and so on) is one more than a multiple of 9. So a number leaves the same remainder as its digit sum when divided by 9 or by 3. For 4 and 8 only the last two or three digits matter, because 100 divides by 4 and 1,000 divides by 8.

Techniques

Add the digits

The remainder by 9 or by 3.

Add the digits. If the sum has more than one digit, add its digits too.
By 9: the single digit you end with is the remainder, except that 9 means remainder 0.
By 3: divide the digit sum by 3 and keep the remainder.

worked example

Example: What is the remainder when 47,582 is divided by 9?

4 + 7 + 5 + 8 + 2 = 26.
2 + 6 = 8.
So 47,582 leaves remainder 8.

Answer: 8

Look at the last digits

Testing whether 4, 6, 8 or 11 divides a number.

By 4: test the last two digits. By 8: test the last three.
By 6: the number must be even and pass the test for 3.
By 11: alternately add and subtract the digits. The result must be 0 or a multiple of 11.

worked example

Example: What is the remainder when 5,318 is divided by 4?

100 divides by 4, so only the last two digits matter: 18.
18 = 4 × 4 + 2.
The remainder is 2, so 4 does not divide 5,318.

Answer: 2

Add one to each exponent

Counting all the positive divisors of a number.

Write the prime factorization, like 72 is 2³ × 3².
Add 1 to each exponent: each prime can appear from 0 times up to its exponent.
Multiply those numbers together.

worked example

Example: How many positive divisors does 60 have?

60 is 2² × 3 × 5.
The exponents 2, 1 and 1 become 3, 2 and 2.
3 × 2 × 2 = 12 divisors.

Answer: 12

Tips by skill

TipRemainder by 9: Add the digits until one digit is left. By 9, that digit is the remainder (9 means 0). By 3, take its remainder by 3.
TipDivisibility tests: Match the test to the divisor: digit sum (3, 9), last two digits (4), last three (8), even and by 3 (6), alternating sum (11).
TipCount the divisors: Factor into primes, add 1 to each exponent, and multiply the results.

Watch out for

Stopping at the digit sum. For 58,643 it is 26, too big to be a remainder by 9. Add again: 2 + 6 is 8.
Checking only that a number is even when testing for 4. 7,226 is even, but 26 is not a multiple of 4.
Multiplying the exponents to count divisors. For 72, which is 2³ × 3², use 4 × 3, not 3 × 2.

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.

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.

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