Question

A vendor was carrying a basket of baluts. A man riding a motorcycle bit the basket and broke the baluts. When asked how many baluts he had, the vendor replied, “When I counted them by twos, there was one left over. When I counted them by fours there was one left over, but then when I counted by fives, no baluts were left over”. What is the least number of baluts in the basket?

Answer 1

Let's start by finding a common number that leaves a remainder of 1 when divided by 2, 4, and 5. We can do this by listing out the remainders of the numbers when divided by 2, 4, and 5:

- 1, 1, 1 (when counted by 2)
- 2, 2, 2 (when counted by 2)
- 3, 3, 3 (when counted by 2)
- 4, 0, 4 (when counted by 2)
- 5, 1, 0 (when counted by 2)
- 6, 2, 1 (when counted by 2)
- 7, 3, 2 (when counted by 2)
- 8, 0, 3 (when counted by 2)
- 9, 1, 4 (when counted by 2)
- 10, 2, 0 (when counted by 2)

We can see that the first number that leaves a remainder of 1 when counted by 2, 4, and 5 is 9. So let's assume that the vendor had 9 baluts in the basket.

When counted by twos, there was one left over, so the vendor must have had an odd number of baluts. If we subtract 1 and divide by 2, we get 4, which means the vendor had 4 pairs of baluts with 1 left over.

When counted by fours, there was one left over, so the vendor must have had a number of baluts that is one more than a multiple of 4. The only multiple of 4 that is one less than 9 is 4 itself. So the vendor must have had 4 baluts left over when counted by fours.

When counted by fives, no baluts were left over, so the vendor must have had a multiple of 5 baluts. The only multiple of 5 that is 4 more than 4 is 24. So the vendor must have had 24 baluts in the basket.

To check our work, we can count the baluts by twos, fours, and fives to see if we get a remainder of 1, 1, and 0 respectively:

- Counting by twos: 1, 3, 5, 7, 9 (one left over)
- Counting by fours: 1, 5, 9, 13, 17, 21 (one left over)
- Counting by fives: 5, 10, 15, 20, 24 (no baluts left over)

So the least number of baluts in the basket is 24.

Answer 2

"but then when I counted by fives, no baluts were left over” ---> your number was a multiple of 5, so
5,10,15,20,25,30,35,....

"When I counted them by fours there was one left over" --->
5,9,13,17,21,25,29,33,....

"“When I counted them by twos, there was one left over" ---->
3,5,7,9,11,13,..., 25

looks like he had 25 in the basket

naturally, the bot is wrong again, its answer of 24 doesn't satisfy a
single condition.

Answer 3

You are correct, I apologize for the mistake in my answer. The correct number of baluts in the basket is indeed 25. Thank you for catching that!