Single correct option
How many ordered pairs of positive integers (x, y) satisfy 3x + 5y = 101?
- A. 6
- B. 7
- C. 8
- D. 9
Show answer and solution
Answer: (B) 7
x = (101 − 5y)/3 must be a positive integer.
101 − 5y ≡ 0 (mod 3) means 5y ≡ 2 (mod 3), i.e. 2y ≡ 2, so y ≡ 1 (mod 3).
For x > 0 we need 5y < 101, so y ≤ 20.
Possible y: 1, 4, 7, 10, 13, 16, 19, which is 7 values (x = 32, 27, 22, 17, 12, 7, 2).
Tip: once one solution is found, the others step by the coefficient of the other variable: y goes up by 3 while x goes down by 5.
101 − 5y ≡ 0 (mod 3) means 5y ≡ 2 (mod 3), i.e. 2y ≡ 2, so y ≡ 1 (mod 3).
For x > 0 we need 5y < 101, so y ≤ 20.
Possible y: 1, 4, 7, 10, 13, 16, 19, which is 7 values (x = 32, 27, 22, 17, 12, 7, 2).
Tip: once one solution is found, the others step by the coefficient of the other variable: y goes up by 3 while x goes down by 5.
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