Question

A bake shop is making boxes of cookies for a party. They put the cookies into boxes of two different sizes. When they put the cookies into boxes of 10, they have 3 left over. When they put the cookies into boxes of 8, they have 1 left over. How many cookies can the bake shop make?

Answer 1

The bake shop can make a batch of 33 cookies, as this number satisfies both conditions of having 3 cookies left over when boxed in groups of 10 and 1 cookie left over when boxed in groups of 8.

Step-by-step explanation:

The problem presented is a classic example of finding the least common multiple (LCM) that satisfies a set of modular arithmetic conditions. The bake shop's situation can be formulated as follows: the total number of cookies, when divided by 10, leaves a remainder of 3, and when divided by 8, leaves a remainder of 1. To solve this, we look for a number that meets both conditions.

To find the smallest number of cookies that can be divided into boxes of 10 with 3 left over (10n + 3) and boxes of 8 with 1 left over (8m + 1), we look for a number that fulfills both equations. By checking multiples of the larger number (10 in this case), we see that 23 satisfies both conditions (as 23 % 10 equals 3 and 23 % 8 equals 7, but the next multiple of 10 plus 3, which is 33, satisfies both conditions because 33 % 10 equals 3 and 33 % 8 equals 1).

Therefore, the bake shop can make a batch of 33 cookies, as it satisfies both conditions of the problem.

Answer 2

The smallest number of cookies the bake shop can make to satisfy the given conditions is 33; this is determined using concepts from number theory and modular arithmetic.

Step-by-step explanation:

The question involves solving a problem related to number theory and modular arithmetic, which falls under the subject of mathematics. Specifically, we are looking to find the smallest number of cookies such that when divided by 10, there are 3 remaining and when divided by 8, there is 1 remaining. This is also known as finding the least common number that satisfies both conditions of the given congruences.

To solve this, we can use the Chinese Remainder Theorem or simply list multiples of each size box until we find a common number that satisfies both conditions. Multiples of 10 with 3 added would be 13, 23, 33, 43, and so on. Multiples of 8 with 1 added would be 9, 17, 25, 33, etc.

The smallest number that appears in both lists is 33. Therefore, the bake shop can make 33 cookies in order to fill boxes completely either way with the respective leftovers.