Numerical answer
How many ordered pairs of positive integers (x, y) satisfy 3x + 7y = 150?
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Answer: 7
We need 150 − 7y to be a positive multiple of 3.
150 ≡ 0 (mod 3) and 7y ≡ y (mod 3), so y ≡ 0 (mod 3).
y = 3 gives x = 43. Further solutions come by adding 3 to y and subtracting 7 from x: (43, 3), (36, 6), (29, 9), (22, 12), (15, 15), (8, 18), (1, 21). The next value of y (24) would make x = −6, not allowed.
That is 7 ordered pairs.
150 ≡ 0 (mod 3) and 7y ≡ y (mod 3), so y ≡ 0 (mod 3).
y = 3 gives x = 43. Further solutions come by adding 3 to y and subtracting 7 from x: (43, 3), (36, 6), (29, 9), (22, 12), (15, 15), (8, 18), (1, 21). The next value of y (24) would make x = −6, not allowed.
That is 7 ordered pairs.
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