Solution to A Basket Full of Eggs

The fewest number of eggs is 300.

300 has an integer quotient when divided by 2 (150), 3 (100), 4 (75), 5 (60), 6 (50), and 301 divided by 7 equals 43.

How do you get the answer? Well, I think there must be a better way, but here's what I did.

I started by finding an easy solution, 2x3x4x5x6. This equals 720, and 721 divided by 7 equals 103. So I knew the basket had no more than 720 eggs.

For a minute let's assume the solution is a 2 digit number, AB. To have an integer quotient when divided by 5 the one's digit (B) must be either 0 or 5. But to have an integer quotient when divided by 2 the one's digit must be either 0, 2, 4, 6 or 8. Thus the one's digit must be 0.

Next, recall that the number of eggs plus 1 has an integer quotient when divided by 7. Well the only 2 digit numbers, with the one's digit being 1, with integer quotients when divided by 7 are 21 and 91. Well, 20 doesn't have an integer quotient when divided by 3 or 6, and 90 doesn't when divided by 4. So the solution must be a 3 digit number, ABC.

Using the same idea as above, I looked for 3 digit numbers, less than 720, with the one's digit being a 1. Again, there are only a few; 161, 231, 301, 371, 441, 511, 581, 651. When you try these you find 160 doesn't have an integer quotient when divided by 3 or 6 and 230 doesn't when divided by 4 or 6. However, 300 does have an integer quotient for 2, 3, 4, 5 and 6, and 301 has an integer quotient when divided by 7!

Let me know if you have a solution with less trial and error.

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