1958 AMC 12 Problem 32
Attempt Problem 32 of the 1958 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1958 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
32.
With a rancher is to buy steers at each and cows at each. If the number of steers and the number of cows are both positive integers, then:
this problem has no solution
there are two solutions with exceeding
there are two solutions with exceeding
there is one solution with exceeding
there is one solution with exceeding
Answer: E
Small Hint:
Write and reduce it modulo
Big Hint:
The congruence and positivity leave only one possible positive value of
Solution:
The purchase equation is Reducing modulo gives Since is positive and the only possibility is Then There is one solution, and
Thus, the correct answer is E.
Problem 32 in Other Years
1950 AMC 12 · 1951 AMC 12 · 1952 AMC 12 · 1953 AMC 12 · 1954 AMC 12 · 1955 AMC 12 · 1956 AMC 12 · 1957 AMC 12 · 1959 AMC 12 · 1960 AMC 12 · 1961 AMC 12 · 1962 AMC 12 · 1963 AMC 12 · 1964 AMC 12 · 1965 AMC 12 · 1966 AMC 12 · 1967 AMC 12 · 1968 AMC 12 · 1969 AMC 12 · 1970 AMC 12 · 1971 AMC 12 · 1972 AMC 12 · 1973 AMC 12