Question

one egg remains when the eggs are removed from the basket 2, 3, 4, 5, or 6 at a time. but no eggs remain if they are removed 7 at a time. find the smallest number of eggs that could have been in the basket using theory of congruence.

Answer 1

Using the Chinese Remainder Theorem, we find that the smallest number of eggs satisfying the given conditions is 120, where the least common multiple (LCM) of 2, 3, 4, 5, and 6 is 60, and 119 (one less than 120) is not divisible by 7.

Step-by-step explanation:

The question involves finding the smallest number of eggs in a basket given certain divisibility conditions. This problem is best tackled using the Chinese Remainder Theorem (CRT), which is a part of number theory in mathematics. We want to find the smallest positive integer n such that:

• n ≡ 1 (mod 2)
• n ≡ 1 (mod 3)
• n ≡ 1 (mod 4)
• n ≡ 1 (mod 5)
• n ≡ 1 (mod 6)
• n ≠ 0 (mod 7)

The conditions n ≡ 1 (mod 2), (mod 3), (mod 4), and (mod 5) mean that n minus 1 is divisible by each of these numbers; however, for the case of (mod 6), since 6 is a multiple of 2 and 3, it is already covered by the previous conditions. The condition n ≠ 0 (mod 7) means n is not divisible by 7. To find the smallest n that satisfies all these conditions simultaneously, we have to look at the multiples of the least common multiple (LCM) of 2, 3, 4, 5, and 6, which is 60, and find the first such number for which when we add 1, it is not divisible by 7. The smallest such number is 119, which is 2*60-1. Therefore, the smallest number of eggs that could have been in the basket is 120.