Divisibility
Quick tests for divisibility, and counting divisors.
Best in your head · no pen, no calculator why?
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the idea
The digits of a number can tell you what a division leaves, without doing the division.
more of the idea
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
- 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.
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
- 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.
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
- 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.
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
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.
practice
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Remainder by 9
worked example
What is the remainder when 90,117 is divided by 3?
- Digit sum: 9 + 0 + 1 + 1 + 7 = 18.
- A number leaves the same remainder by 3 as its digit sum: 18 = 3 × 6 + 0, so the remainder is 0.
Answer: 0
Divisibility tests
worked example
Which number is divisible by 6?
- 9,400
- 2,088
- 1,821
- 4,502
- 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.
- Each of the others fails one of the two tests.
- So the answer is 2,088.
Answer: 2,088
Count the divisors
worked example
How many positive divisors does 63 have?
- 63 = 3² × 7.
- A divisor uses one power of each prime: 3⁰ to 3² (3 choices) and 7⁰ to 7¹ (2 choices).
- 3 × 2 = 6 divisors.
Answer: 6