# Problem of the Day

Melissa enters a charity raffle in which bags of candy are being given away as prizes. She will be randomly assigned a number from the set of positive integers from 1 to 500 and receive a bag filled with that many pieces of candy. She decides that if the number of pieces of candy she receives is such that she can distribute the candy equally among her three children OR such that she can distribute the candy equally among her children, herself, and her husband, then she will keep it. In all other cases, she will give the candy to her neighbor if only if it can be distributed equally among the seven members of her neighbor’s family. What is the probability that Melissa gives the bag of candy to her neighbor?

A. 0.068

B. 0.076

C. 0.096

D. 0.114

E. 0.142

### Solution

[latexpage]

Melissa will give the bag of candy to the neighbor if the number of pieces is a multiple of 7 but not a multiple of 5 or 3. There are $\lfloor\frac{500}{7}\rfloor=71$ multiples of 7 less than 500. Of these,Â Â $\lfloor\frac{500}{35}\rfloor=14$ are also multiples of 5, andÂ Â $\lfloor\frac{500}{21}\rfloor=23$ are also multiples of 3. $71-14-23=34$, but this subtracts out multiples of $3\cdot5\cdot7=105$ twice, so add $\lfloor\frac{500}{105}\rfloor=4$ back in to get 38. So, 38 of the 500 possible numbers would result in her giving the candy to her neighbor. $\frac{38}{500}=\frac{76}{1000}=0.076$

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