1985 AIME Problem 7

Attempt Problem 7 of the 1985 AIME 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 1985 AIME solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Assume that

a,

b,

c,

and

d

are positive integers such that

a^5=b^4,

c^3=d^2,

and

c-a=19.

Determine

d-b.

Answer

Parametrize the solutions of

a^5=b^4

and

c^3=d^2

Factor the resulting difference

s^2-t^4

Comparing prime exponents, write

\begin{aligned} a&=t^4,\quad b=t^5,\\ c&=s^2,\quad d=s^3 \end{aligned}

for positive integers

s,

t.

Then

(s-t^2)(s+t^2)=s^2-t^4=19.

Since

19

is prime, the factors are

1

and

19,

giving

s=10

and

t^2=9.

Thus

t=3,

and

\begin{aligned} d-b&=10^3-3^5\\ &=1000-243=757. \end{aligned}

Problem 7 in other years