1967 AMC 12 Problem 22

Attempt Problem 22 of the 1967 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 1967 AMC 12 solutions, or check the answer key.

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

22.

For natural numbers, when

P

is divided by

D,

the quotient is

Q

and the remainder is

R.

When

Q

is divided by

D',

the quotient is

Q'

and the remainder is

R'.

Then, when

P

is divided by

DD',

the remainder is:

R+R'D

R'+RD

RR'

R

R'

Write

P=QD+R

and

Q=Q'D'+R'

Substitute the second equation into the first and isolate the multiple of

DD'

Substitution gives

\begin{aligned} P&=(Q'D'+R')D+R\\ &=Q'(DD')+(R'D+R). \end{aligned}

The remainder bounds imply

R'D+R\lt DD',

so this is indeed the remainder.

Therefore, the correct answer is A.

Problem 22 in other years