Question 1

If four different positive integers $$m, n, p, q$$ satisfy the equation $$(7-m)(7-n)(7-p)(7-q) = 4$$ then the sum $$m+n+p+q$$ is equal to

Solution

The four numbers $$7-m, 7-n, 7-p, 7-q$$ are distinct integers whose product is 4, so they must be $$1, -1, 2, -2$$ in some order. This gives $$m, n, p, q$$ equal to $$6, 8, 5, 9$$ in some order, and all four are different positive integers as required. Hence $$m+n+p+q = 6+8+5+9 = 28$$.

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If four different positive integers $$m, n, p, q$$ satisfy the equation $$(7-m)(7-n)(7-p)(7-q) = 4$$ then the sum $$m+n+p+q$$ is equal to

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