We are doing GCF and lcm and prime factorization.
14 pictures of family, 8 of pets. 5 of both. How many pictures in all
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Draw a picture of two intersecting circles. The left circle stands for the family, the right for the pets. Their intersection contains 5, the pictures of both. The total in left circle is 14, or 9 + 5 in the intersection. The right circle total is 8. 5 in the intersection plus 3 in the circle outside the intersection.
The total number of pictures is 9+5+3=17.
I have no understanding of the reason for your remark about GCF, LCM, and prime factorization, but they are certainly good to know.
I like to do these kinds of problems with a Venn diagram. In this case there are two overlapping circles. The common area represents the pictures that are both family and pets of which there are 5. You wee also given that there are 14 family pictures so there must be 14-5 or 9 pictures of family alone. With pets there are 8 total and therefor there must be 8-5 or 3 pictures of only pets.
Adding the pictures gives 9+3+5 = 17 pictures in all.
Another quicker way using set notation is F∪P = F + P - F∩P =14 + 8 - 5 =17