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

5

Expert Reply

15

Bookmarks

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: **4806**

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

6

Expert Reply

1

Bookmarks

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

_________________

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

Re: QOTD#13 The integer m is a multiple of 154, 250, and 264.
[#permalink]
26 Nov 2021, 05:02

I think there's an error with the solution here, please correct me if I'm wrong.

If we consider m to be a product of 154, 250 and 264 i.e. 10164000.

It'd still be considered a multiple of all 3 numbers and in that case 176 and 242 are perfectly dividing 10164000, but 2500 isn't. So then the answer should be only Option E.

Posted from my mobile device

If we consider m to be a product of 154, 250 and 264 i.e. 10164000.

It'd still be considered a multiple of all 3 numbers and in that case 176 and 242 are perfectly dividing 10164000, but 2500 isn't. So then the answer should be only Option E.

Posted from my mobile device

Re: QOTD#13 The integer m is a multiple of 154, 250, and 264.
[#permalink]
26 Nov 2021, 06:47

achal wrote:

I think there's an error with the solution here, please correct me if I'm wrong.

If we consider m to be a product of 154, 250 and 264 i.e. 10164000.

It'd still be considered a multiple of all 3 numbers and in that case 176 and 242 are perfectly dividing 10164000, but 2500 isn't. So then the answer should be only Option E.

Posted from my mobile device

If we consider m to be a product of 154, 250 and 264 i.e. 10164000.

It'd still be considered a multiple of all 3 numbers and in that case 176 and 242 are perfectly dividing 10164000, but 2500 isn't. So then the answer should be only Option E.

Posted from my mobile device

First off, 2500 is a divisor of 10164000. So, with your approach, there are no correct answers.

Also, multiplying all three values together will yield incorrect conclusions.

To see why, let's say the three numbers are 2, 3 and 6.

If we multiply them all together we get m = 36, which means 18 must be a factor of m.

However, m could also equal 6 (a much smaller number than 36). In this case, 18 is NOT a factor of m.

_________________

Re: QOTD#13 The integer m is a multiple of 154, 250, and 264.
[#permalink]
08 May 2023, 09:46

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