Level 2 · Ages 8–11
Read it again any time, including while you work through the exercises.
You have 12 sweets and you want to share them fairly. With 3 friends and you, everyone gets 3: a perfect share! With 5 people, it is impossible without cutting a sweet. We say that 4 divides 12, but 5 does not divide 12.
Two magic words go together. The multiples of 6 are the results of the six times table: 6, 12, 18, 24, 30... and it never stops. The divisors of 12 are the numbers that share 12 with no remainder: 1, 2, 3, 4, 6 and 12. The two ideas are linked: saying 12 is a multiple of 4 is exactly saying 4 is a divisor of 12, because 4 × 3 = 12.
What about big numbers? No need to compute everything, there are detective tricks called divisibility rules. A number is divisible by 2 if it ends in 0, 2, 4, 6 or 8. By 5 if it ends in 0 or 5. By 10 if it ends in 0. And the most surprising: by 3 if the sum of its digits is in the three times table, and by 9 if the sum of its digits is in the nine times table.
Example: is 234 divisible by 3? I add the digits: 2 + 3 + 4 = 9. And 9 is in the three times table, so yes, 234 is divisible by 3, without doing a single division!
In this lesson, you will learn to spot multiples and divisors in the blink of an eye. The investigation begins!
Is 8 a divisor of 56?
8 × ? = 56
I check whether 56 is in the eight times table: is there a whole number of groups of 8 that makes 56?
8 × 7 = 56
Yes! 7 groups of 8 make exactly 56, with no remainder.
56 = 8 × 7
So 8 divides 56. And I learn two things at once: 7 is also a divisor of 56, and 56 is a multiple of both 8 and 7.
Result: Yes, 8 is a divisor of 56, because 8 × 7 = 56.
Without doing any division, show that 234 is divisible by 3.
2 + 3 + 4 = 9
I use the divisibility rule for 3: I add the digits of 234.
9 = 3 × 3
The sum is 9, and 9 is in the three times table.
234 = 3 × 78
The rule guarantees that 234 is divisible by 3. Check: 3 × 78 = 234, exactly right!
Result: 234 is divisible by 3, because 2 + 3 + 4 = 9 and 9 is a multiple of 3.