2021 AMC 12A Fall Problem 8

Question

Let

M

be the least common multiple of all the integers

10

through

30,

inclusive. Let

N

be the least common multiple of

M,

32,

33,

34,

35,

36,

37,

38,

39,

and

40.

What is the value of

\dfrac{N}{M}?

1

2

37

74

2886

M

already contains

2^4, 3^3, 5^2, 7,

and all primes up to

29;

check what

32

–

40

add

32 = 2^5

raises the power of

2,

and

37

is a new prime; nothing else is new

M = \operatorname{lcm}(10, \ldots, 30)

contains

2^4

(from

16

),

3^3

(from

27

),

5^2

(from

25

),

7,

and every prime up to

29.

Among

32, \ldots, 40,

the only new contributions are

32 = 2^5,

which raises the power of

2

from

2^4

to

2^5,

and the new prime

37.

Everything else factors into primes and powers already in

M.

Therefore

\dfrac{N}{M} = 2 \cdot 37 = 74.

Thus, the correct answer is D.

Problem 8 in other years