1969 AMC 12 Problem 24

Attempt Problem 24 of the 1969 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 1969 AMC 12 solutions, or check the answer key.

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24.

When the natural numbers

P

and

P',

with

P\gt P',

are divided by the natural number

D,

the remainders are

R

and

R',

respectively. When

PP'

and

RR'

are divided by

D,

the remainders are

r

and

r',

respectively. Then:

r\gt r'

always

r\lt r'

always

r\gt r'

sometimes, and

r\lt r'

sometimes

r\gt r'

sometimes, and

r=r'

sometimes

r=r'

always

Write

P\equiv R\pmod D

and

P'\equiv R'\pmod D

Multiply the two congruences

By definition of the remainders,

\begin{aligned} P&\equiv R\pmod D,\\ P'&\equiv R'\pmod D. \end{aligned}

Multiplying gives

PP'\equiv RR'\pmod D.

Since each has a unique remainder between

0

and

D-1,

their remainders must be equal:

r=r'.

Therefore, the correct answer is E.

Problem 24 in other years