1971 AMC 12 Problem 14

Attempt Problem 14 of the 1971 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 1971 AMC 12 solutions, or check the answer key.

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

14.

The number

2^{48}-1

is exactly divisible by two numbers between

60

and

70.

These numbers are:

61,

63

61,

65

63,

65

63,

67

67,

69

Use the fact that

2^m-1

divides

2^n-1

whenever

n

is divisible by

m

Recognize

63=2^6-1

and that

2^{12}-1

is divisible by

65

Because

48

is divisible by

6,

the number

2^{48}-1

is divisible by

63=2^6-1.

Also

2^{12}-1=4095=63\cdot65,

and

48

is divisible by

12,

2^{48}-1

is divisible by

65.

Thus the two numbers are

63

and

65.

Therefore, the correct answer is C.

Problem 14 in other years