A number from its remainders, step by step
A number leaves 2 when divided by 3, 3 when divided by 5, and 2 when divided by 7.
What is the smallest number of marbles there could be?
Climb one list until it fits the others
- Nothing is given but leftovers: divided by 5 it leaves 1 and divided by 8 it leaves 4. No single one of those picks out a number, and together they pick out exactly one small one.
- Take the conditions one at a time, starting with the first. The numbers that leave 1 when divided by 5 are 1, 6, 11, 16, 21, 26 and so on — they go up in 5s, because adding 5 cannot change what a division by 5 leaves over.
- Now walk up that list in 5s until one of them also leaves 4 when divided by 8: 1, 6, 11, 16, … do not, and 36 does.
- So the smallest is 36 marbles. Check it against every condition rather than the last one: 5 × 7 = 35, leaving 1 and 8 × 4 = 32, leaving 4.
How it works.
- 01
Nothing is given but leftovers: divided by 5 it leaves 1 and divided by 8 it leaves 4. No single one of those picks out a number, and together they pick out exactly one small one.
- 02
Take the conditions one at a time, starting with the first. The numbers that leave 1 when divided by 5 are 1, 6, 11, 16, 21, 26 and so on — they go up in 5s, because adding 5 cannot change what a division by 5 leaves over.
- 03
Now walk up that list in 5s until one of them also leaves 4 when divided by 8: 1, 6, 11, 16, … do not, and 36 does.
- 04
So the smallest is 36 marbles. Check it against every condition rather than the last one: 5 × 7 = 35, leaving 1 and 8 × 4 = 32, leaving 4.