Let a and b be integers. If the set of positive integer values of x satisfying both inequalities 9x - a ≥ 0 and 8x - b

Question

Let a and b be integers. If the set of positive integer values of x satisfying both inequalities 9x - a ≥ 0 and 8x - b < 0 is exactly {1, 2, 3}, then how many ordered pairs (a, b) are possible?

Accepted Answer

72. x>=a/9 and x<b/8 must give exactly {1,2,3}. Include x=1 and exclude lower bound: 0<a/9<=1 so a in {1,...,9} (9 values). Include 3, exclude 4: 3<b/8<=4 so 24<b<=32, b in {25,...,32} (8 values). Total ordered pairs = 9x8 = 72.

Let a and b be integers. If the set of positive integer values of x satisfying both inequalities 9x - a ≥ 0 and 8x - b < 0 is exactly {1, 2, 3}, then how many ordered pairs (a, b) are possible?

Solution

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