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# Counting

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Andrew chooses a number from 1 to 100, and Mary also chooses a number from 1 to 100. (They may choose the same number.) It turns out that the product of their numbers is a multiple of 5. In how many ways could Andrew and Mary have chosen their numbers?

Apr 20, 2022

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There are 3 cases:

Mary gets a multiple of 5 but Andrew doesn't: 20 ways for Mary to get a multiple of 5, 80 ways for Andrew to get a number that isn't a multiple of 5

So, there are $$20 \times 80 = 1600$$ ways to do this.

Andrew gets a multiple of 5 but Mary doesn't: 20 ways for Andrew to get a multiple of 5, 80 ways for Mary to get a number that isn't a multiple of 5

So, there are $$20 \times 80 = 1600$$ ways to do this.

Both get a multiple of 5: There are 20 ways for Andrew to get a multiple of 5, 20 ways for Mary to get a multiple of 5

So, there are $$20 \times 20 = 400$$ ways to do this.

Thus, there are $$1600+400+1600 = \text{____}$$ ways.

Apr 20, 2022