Counting on in 6s gives 6, 12, 18, 24, 30, 36, and never ends. Those are the
multiples of 6, each 6 times a whole number, so 6
is a factor of every one of them.
A division decides whether a number is on the list: 24 ÷ 6 = 4 with nothing over, so 24 is a
multiple of 6; 25
÷ 6 = 4 r 1, so 25 is not, and the same goes for any number that leaves a remainder.
The last digit tests 2, 5 and 10
Short division decides whether 4 divides 4,932, but it takes four steps of
working for a yes or a no.
A divisibility test reads the digits and gives the yes or the no without the division; the quotient, when it is wanted, still has to come from
dividing.
Divisible by 2
The last digit is 0, 2, 4, 6 or 8.
The last digit of 4,932 is 2. Yes: 4,932 ÷ 2 = 2,466
Why it works
Every whole number is a multiple of 10 with its last digit added on: 4,932 = 4,930 + 2. A ten is five 2s, so 4,930 is a whole number of 2s and leaves nothing over; only the 2 on the end is left to decide. 2 is one whole 2, so nothing is left over at all.
Change the number to 4,933 and the 4,930 is untouched; 3 is one 2 with 1 left over: 4,933 ÷ 2 = 2,466 r 1. That is the whole test: 0, 2, 4, 6 and 8 are whole numbers of 2s, and the other five digits each leave 1 behind.
Divisible by 5
The last digit is 0 or 5.
The last digit of 4,932 is 2. No: 4,932 ÷ 5 = 986 r
2
Why it works
4,932 = 4,930 + 2, and a ten is two 5s, so 4,930 is a whole number of 5s and leaves nothing over. Only the 2 on the end is left to decide, and 2 is smaller than 5, so the remainder is 2.
Divisible by 10
The last digit is 0.
The last digit of 4,932 is 2. No: 4,932 ÷ 10 = 493 r
2
Why it works
4,932 = 4,930 + 2, and 4,930 is 493 tens, so it divides by 10 exactly. Only the 2 on the end is left to decide, and every digit from 1 to 9 is smaller than 10, so each one survives as the remainder. Only a 0 leaves nothing behind.
The last two digits test divisibility by 4
Divisible by 4
The number made by the last two digits is a multiple of 4.
The last two digits of 4,932 make 32, which is 4
× 8. Yes: 4,932 ÷ 4 = 1,233
Why it works
A hundred is a whole number of 4s, 100 = 4 × 25, so 4,932 is cut after its last two digits: 4,932 = 4,900 + 32. The 4,900 is 49 hundreds and leaves nothing over; only the 32 is left to decide.
One digit is not enough, because a ten is not a whole number of 4s: 10 = 4 × 2 + 2.
The digit sum tests 3 and 9
Divisible by 9
The digits add to a multiple of 9.
Adding the digits of 4,932 gives 4 + 9 + 3 + 2 = 18, and
9 divides 18. Yes: 4,932 ÷ 9 = 548
Why it works
No cut helps for 9, because a ten, a hundred and a thousand each miss a multiple of 9 by 1: 10 = 9 + 1; 100 = 99 + 1; 1,000 = 999 + 1.
So each place of 4,932 splits into a whole number of 9s with its own digit left over: 4,000 = 4 × 999 + 4; 900 = 9 × 99 + 9; 30 = 3 × 9 + 3; and the 2 is a single digit already.
The parts made of 999s, 99s and 9s add to 4 × 999 + 9 × 99 + 3 × 9 = 4,914 = 9 × 546, a multiple of 9 whatever the digits are. The digits left over add to 18 = 9 × 2, two more 9s.
Adding the digits of the digit sum keeps the remainder: 18 is a ten and an 8, the ten is 9 + 1, so 18 and 1 + 8 = 9 leave the same remainder.
Divisible by 3
The digits add to a multiple of 3.
Adding the digits of 4,932 gives 4 + 9 + 3 + 2 = 18, and
3 divides 18. Yes: 4,932 ÷ 3 = 1,644
Why it works
9, 99 and 999 are multiples of 3 as well, 3 × 3, 3 × 33 and 3 × 333, so the split for 9 works for 3 too. The part of 4,932 made of 999s, 99s and 9s is 4,914 = 3 × 1,638, and the digits left behind add to 18 = 3 × 6.
The two tests read the same sum and can answer differently. 4,524 adds to 15, which 3 divides and 9 does not, so 4,524 ÷ 3 = 1,508 while 4,524 ÷ 9 = 502 r 6.
Six tests on one number
Every whole number answers all six tests from its digits alone; the division only confirms what the
digits said.
A whole number with three or four digits.
2The last digit is 5.No: 2,835 ÷ 2 = 1,417 r 1
5The last digit is 5.Yes: 2,835 ÷ 5 = 567
10The last digit is 5.No: 2,835 ÷ 10 = 283 r 5
4The last two digits make 35.No: 2,835 ÷ 4 = 708 r 3
2,835 is a multiple of 3, 5 and 9, and not of 2, 4 or
10.
A whole number with three or four digits is
needed.
Common mistakes
Calling 6 a multiple of 246 is a factor of 24, not a multiple of it: the multiples of 24 are 24, 48, 72 and so on, and 6 is not
among them.
Testing 2 on the digit sumThe digits of 4,527 add to 18, which is even, and 4,527 is odd: 4,527 ÷ 2 = 2,263
r 1. Adding the digits tests 3 and 9; the test for 2 reads the last digit alone.
Testing 4 on the last digit alone734 ends in 4, and 4 is a multiple of 4, but 734 ÷ 4 = 183 r 2: 34 is not a multiple of 4,
while 732 passes because 32 is.
Looking for a 9 instead of adding the digits4,536 contains no digit 9, but its digits add to 18 and 4,536 ÷ 9 = 504. The digit sum,
not the presence of a 9, decides the test.
Summary
The multiples of 6 are 6, 12, 18, 24 and onwards, and 6 is a factor of every one of
them.
The last digit tests divisibility by 2, 5 and 10; the last two digits test divisibility
by 4.
The digit sum tests divisibility by 3 and 9.
A divisibility test gives a yes or a no; division is still needed to find the quotient.