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QOTD#13 The integer m is a multiple of 154, 250, and 264.
[#permalink]
09 Nov 2016, 15:41

3

Expert Reply

6

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Question Stats:

The integer m is a multiple of 154, 250, and 264. Which of the following do NOT have to be

factors of m?

Indicate all possible values

A. 176

B. 242

C. 275

D. 924

E. 2,500

F. 7,000

_________________

factors of m?

Indicate all possible values

A. 176

B. 242

C. 275

D. 924

E. 2,500

F. 7,000

Drill 3

Question: 12

Page: 290

Question: 12

Page: 290

Show: :: OA

A,B,E

_________________

Sandy

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Retired Moderator

Joined: **07 Jun 2014 **

Posts: **4805**

WE:**Business Development (Energy and Utilities)**

Re: QOTD#13 The integer m is a multiple of 154, 250, and 264.
[#permalink]
09 Nov 2016, 15:45

5

Expert Reply

Explanation

To solve this question, turn large numbers into small numbers by working with factors. The prime factors of 154 are 2, 7, and 11; the prime factors of 264 are 2, 2, 2, 3, and 11; and the prime factors of 250 are 2, 5, 5, and 5. The only numbers that must be a factor of m are those made up of factors contained in the other three numbers. You can’t recount factors that overlap in the different numbers, so you know that m is made up of, at least, three 2’s, one 3, three 5’s, one 7, and one 11. Now check the answers. The prime factors of 176 are 2, 2, 2, 2, and 11, which is one 2 too many, so choice (A) is not a factor; since the question asks you to identify which choices are not factors, choice (A) is part of the credited response.

The prime factors of 242 are 2, 11, and 11, which is one 11 too many, so (B) is also not a factor. The prime factors of 275 are 5, 5, and 11, so choice (C) is a factor. The prime factors of 924 are 2, 2, 3, 7, and 11, so choice (D) is a factor. The prime factors of 2,500 are 2, 2, 5, 5, 5, and 5, which is one 5 too many, so choice (E) is not a factor. And, finally, the prime factors of 7,000 are 2, 2, 2, 5, 5, 5, and 7, so choice (F) is a factor. The correct answers are choices (A), (B), and (E).

_________________

Sandy

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To solve this question, turn large numbers into small numbers by working with factors. The prime factors of 154 are 2, 7, and 11; the prime factors of 264 are 2, 2, 2, 3, and 11; and the prime factors of 250 are 2, 5, 5, and 5. The only numbers that must be a factor of m are those made up of factors contained in the other three numbers. You can’t recount factors that overlap in the different numbers, so you know that m is made up of, at least, three 2’s, one 3, three 5’s, one 7, and one 11. Now check the answers. The prime factors of 176 are 2, 2, 2, 2, and 11, which is one 2 too many, so choice (A) is not a factor; since the question asks you to identify which choices are not factors, choice (A) is part of the credited response.

The prime factors of 242 are 2, 11, and 11, which is one 11 too many, so (B) is also not a factor. The prime factors of 275 are 5, 5, and 11, so choice (C) is a factor. The prime factors of 924 are 2, 2, 3, 7, and 11, so choice (D) is a factor. The prime factors of 2,500 are 2, 2, 5, 5, 5, and 5, which is one 5 too many, so choice (E) is not a factor. And, finally, the prime factors of 7,000 are 2, 2, 2, 5, 5, 5, and 7, so choice (F) is a factor. The correct answers are choices (A), (B), and (E).

_________________

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Re: QOTD#13 The integer m is a multiple of 154, 250, and 264.
[#permalink]
15 Oct 2018, 10:56

Is it feasible to work this out in 1'45"?

Re: QOTD#13 The integer m is a multiple of 154, 250, and 264.
[#permalink]
15 Oct 2018, 13:41

1

Expert Reply

yes.

However, the time allotted is on average.

You could take a bit more if the question is daunting and difficult.

Regards

_________________

However, the time allotted is on average.

You could take a bit more if the question is daunting and difficult.

Regards

_________________

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QOTD#13 The integer m is a multiple of 154, 250, and 264.
[#permalink]
06 Oct 2021, 20:07

1

sandy wrote:

The integer m is a multiple of 154, 250, and 264. Which of the following do NOT have to be

factors of m?

Indicate all possible values

A. 176

B. 242

C. 275

D. 924

E. 2,500

F. 7,000

factors of m?

Indicate all possible values

A. 176

B. 242

C. 275

D. 924

E. 2,500

F. 7,000

Drill 3

Question: 12

Page: 290

Question: 12

Page: 290

Show: :: OA

A,B,E

A very straight forward answer.

if m is a multiple of 154, 250, & 256, it means that m is divisible by 154, 250, 264. So;

\(m/154\), \(m/ 250\), \(m/264\) will yield a remainder of zero.

This means that all the prime factors of each of the denominators are in the numerator m.

Prime factors of 154 = 2 * 7 * 11

Prime factors of 250 = 2 * 5 * 5 * 5

Prime factors of 264 = 2 * 2 * 2 * 3 * 11

So, m at least have the following prime factors.

Three 2s, one 3, three 5s, one 7, and one 11.

Now check each option.

Option A: prime factors of 176 -> 2,2,2,2,11 ----> We do not have four 2s in m. Therefore 176 can't be a factor of m. Correct

Option B: prime factors of 242 -> 2,11,11 ----> Not a factor. Correct

Option C: prime factors of 275 -> 5,5,11 ---> Factor. Incorrect

Option D: prime factors of 924 -> 2,2,3,7,11 ---> Factor. Incorrect

Option E: prime factors of 2500 -> 2,2,5,5,5,5 ---> Not Factor. m doesn't have four 5s. Correct

Option F: prime factors of 7000 -> 2,2,2,5,5,5,7 ---> Factor. Incorrect

Show: ::

Answer: A, B, E

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