Answer:
301
Explanation:
You want a number that has a remainder of 1 when divided by 2, 3, 4, 5, or 6, and has a remainder of 0 when divided by 7.
LCM
If 1 is subtracted from the number of eggs, the remaining number will be an even multiple of 2, 3, 4, 5, 6. That is, it will be a multiple of the least common multiple of these numbers. That LCM will be ...
LCM(LCM(6, 5), 4) . . . . . . 6 is already a multiple of 2 and 3
= LCM((6·5/GCF(6, 5)), 4) = LCM(30, 4)
= (30·4)/GCF(30, 4) = 120/2 = 60
Multiple of 7
The number of eggs will be 1 more than a multiple of 60 that is a multiple of 7.
We only need to try multiples of 60 up to 7×60. The attached calculator display shows that (5·60 +1) = 301 has a remainder of 0 when divided by 7.
The number of eggs in the cart is 301.
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Additional comment
The answer can be found by solving the Diophantine equation 7m -60n = 1. Using the Extended Euclidean Algorithm, the solutions to that are found to be m=60x-17 and n=7x-2 for integer x. The value x=1 gives (m, n) = (43, 5), or 301 -300 = 1. The next higher value for 7m is 721.