A farmer has a flock of around 400 sheep. Keen to find out the exact number, he started counting. He counted them in batches, hoping to save time, but lost track at the end anyway. All he remembers is that when he counted them three at a time, he had one left over. When he counted them four at a time he had two left over. When he counted them five at a time he had four left over. And when he counted them seven at a time, he had two left over.
Can you work out how many sheep he has?
Hints
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"When he counted them five at a time, he had four left over."
This means that if you divided the total number of sheep by five, you'd have four left over. Think about which numbers produce a remainder of four when you divide them by five and you'll see that they all end with a 4 or a 9. The total number of sheep in the farmer's flock must end with 4 or 9.
"When he counted them four at a time, he had two left over."
This means that if you divided the total number of sheep by four, this leaves you with a remainder of two. Think about which numbers produce a remainder of two, and you may notice that they're all even numbers. This fact, along with what Hint 1 told you, means that the number of sheep in the farmer's flock must end with a 4.
"When he counted them seven at a time, he had two left over."
This means that if you divided the total number of sheep by seven, this leaves you with a remainder of two. Bear in mind that the framer has around 400 sheep. The only numbers that can be divided by seven with a remainder of two, and that end with a 4 (as established in Hints 1 and 2) are 324, 394 and 464.
Hint 3 gave you three possible answers: 324, 394 or 464.
Try dividing 464 by four. What happens?
And now for 324. See what happens when you divide it by 3 and 4.
You should have the answer now!
Solution
Incorrect
Too bad.
Make sure you haven't made a mistake with the number of sheep left over in each case.
Correct
Shear class!
The farmer has 394 sheep. Six more and he'd have exactly 400!