Sarah's mom made cookies for Sarah and her four friends. There's one cookie too many to divide evenly among the friends. Then two more friends showed up. Sarah gave her mom one cookie, which allowed Sarah to divide all the cookies evenly. How many cookies did Sarah's mom originally make?

36

12 To find out how many cookies Sarah's mom originally made, we can use a simple algebraic approach.

Let's assume that Sarah's mom originally made 'x' cookies. According to the problem, we know that there is one cookie too many to divide evenly among Sarah and her four friends. So, if we divide 'x' cookies among the five original friends, each friend would get (x / 5) cookies.

Now, the problem states that two more friends showed up. So, Sarah gave her mom one cookie. With this exchange, the total number of cookies becomes (x - 1). Further, since there are now seven friends in total, if we divide the remaining cookies evenly, each friend would get ((x - 1) / 7) cookies.

According to the problem, Sarah giving her mom one cookie allowed her to divide the cookies evenly among all the friends. This means that ((x - 1) / 7) should be a whole number. In other words, ((x - 1) / 7) should be divisible by 1 without any remainder.

Let's set up the equation to solve for 'x':

((x - 1) / 7) = whole number

If we increase 'x' by 1, the numerator of the fraction increases, but since the numerator should be divisible by 7, 'x - 1' must be a multiple of 7.

Let's try different values of x - 1 that are multiples of 7 until we find a whole number solution:

If x - 1 = 7, then ((x - 1) / 7) = (7 / 7) = 1 (a whole number)
If x - 1 = 14, then ((x - 1) / 7) = (14 / 7) = 2 (a whole number)

So, the smallest value that satisfies the equation is x - 1 = 7. Therefore, Sarah's mom originally made 7 + 1 = 8 cookies.

Hence, Sarah's mom originally made 8 cookies.