1958 AMC 12 Problem 32

Question

With

\$1000

a rancher is to buy steers at

\$25

each and cows at

\$26

each. If the number of steers

s

and the number of cows

c

are both positive integers, then:

this problem has no solution

there are two solutions with

s

exceeding

c

there are two solutions with

c

exceeding

s

there is one solution with

s

exceeding

c

there is one solution with

c

exceeding

s

Write

25s+26c=1000

and reduce it modulo

25

The congruence and positivity leave only one possible positive value of

c

The purchase equation is

25s+26c=1000.

Reducing modulo

25

gives

c\equiv0\pmod{25}.

Since

c

is positive and

26c<1000,

the only possibility is

c=25.

Then

s=\frac{1000-26(25)}{25}=14.

There is one solution, and

c>s.

Thus, the correct answer is E.

Problem 32 in other years