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# X is the least common multiple of 96, 196, 300, which number below is

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Joined: 02 Sep 2009
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X is the least common multiple of 96, 196, 300, which number below is  [#permalink]

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01 Nov 2019, 05:30
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55% (hard)

Question Stats:

65% (02:16) correct 35% (02:21) wrong based on 80 sessions

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X is the least common multiple of 96, 196, 300, which number below is not factor of X?

A. 600
B. 700
C. 900
D. 2100
E. 4900

Are You Up For the Challenge: 700 Level Questions: 700 Level Questions

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Re: X is the least common multiple of 96, 196, 300, which number below is  [#permalink]

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01 Nov 2019, 06:07
1
Bunuel wrote:
X is the least common multiple of 96, 196, 300, which number below is not factor of X?

A. 600
B. 700
C. 900
D. 2100
E. 4900

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96 = 2^5*3
196 = 2^2*7^2
300 = 2^2*3*5^2

X = LCM(96, 196, 300) = 2^5*3*5^2*7^2

Since 900 = 2^2*3^2*5^2 and X has only 3^1 as a factor, 900 cannot be the factor of X

IMO Option C

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Re: X is the least common multiple of 96, 196, 300, which number below is  [#permalink]

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01 Nov 2019, 07:13
Bunuel wrote:
X is the least common multiple of 96, 196, 300, which number below is not factor of X?

A. 600
B. 700
C. 900
D. 2100
E. 4900

Are You Up For the Challenge: 700 Level Questions: 700 Level Questions

LCM of 96, 196, 300 = 2^5*7^2*3*5^2
out of given options C ; 900 is not factor of 2^5*7^2*3*5^2
IMO C
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Re: X is the least common multiple of 96, 196, 300, which number below is  [#permalink]

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01 Nov 2019, 07:50
Top Contributor
Bunuel wrote:
X is the least common multiple of 96, 196, 300, which number below is not factor of X?

A. 600
B. 700
C. 900
D. 2100
E. 4900

Are You Up For the Challenge: 700 Level Questions: 700 Level Questions

In case some students are wondering how we find the LCM of 96. 196 and 300, here's a step-by-step solution:

If X is a multiple of 96, then X = 96k (where k is some positive integer)
In other words, X = (2)(2)(2)(2)(2)(3)(k)

196 = (2)(2)(7)(7)
Since X is a multiple of 196, X needs two 2's and two 7's in its prime factorization.
We already know that X = (2)(2)(2)(2)(2)(3)(k), so we already have the two 2's. BUT, we need to add two 7's to the product.
We get: X = (2)(2)(2)(2)(2)(3)(7)(7)(k)

300 = (2)(2)(3)(5)(5)
Since X is a multiple of 300, then X needs two 2's, one 3 and two 5's in its prime factorization.
We already know that X = (2)(2)(2)(2)(2)(3)(7)(7)(k), so we already have the two 2's, and the one 3. BUT, we need to add two 5's to the product.
We get: X = (2)(2)(2)(2)(2)(3)(5)(5)(7)(7)(k)

So, the very smallest value of X = (2)(2)(2)(2)(2)(3)(5)(5)(7)(7)

When we get to C (900), we can see that X must be divisible by 9 (Since 9 is a factor of 900)
For X to be divisible by 9, X must have two 3's in its prime factorization.
However, we do NOT have two 3's in (2)(2)(2)(2)(2)(3)(5)(5)(7)(7)

Cheers,
Brent
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Re: X is the least common multiple of 96, 196, 300, which number below is  [#permalink]

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19 Nov 2019, 01:42
Bunuel wrote:
X is the least common multiple of 96, 196, 300, which number below is not factor of X?

A. 600
B. 700
C. 900
D. 2100
E. 4900

Are You Up For the Challenge: 700 Level Questions: 700 Level Questions

96 = 2^5 * 3
196 = 2^2 * 7^2
300 = 2^2 * 3 * 5^2

LCM(96,196,300) = X = 2^5*3*5^2*7^2

From the options, we can see that C = 900 which has 3^2 but maximum power of 3 in X is 1. so 900 cannot be a factor of X.
IMO - C
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Re: X is the least common multiple of 96, 196, 300, which number below is  [#permalink]

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22 Nov 2019, 13:11
Bunuel wrote:
X is the least common multiple of 96, 196, 300, which number below is not factor of X?

A. 600
B. 700
C. 900
D. 2100
E. 4900

Are You Up For the Challenge: 700 Level Questions: 700 Level Questions

Breaking each number into prime factors, we have:

96 = 32 x 3 = 2^5 x 3^1

196 = 4 x 49 = 2^2 x 7^2

300 = 3 x 100 = 2^2 x 3^1 x 5^2

So the LCM is 2^5 x 3^1 x 5^2 x 7^2.

Since 900 = 2^2 x 3^2 x 5^2, it can’t be a factor of X because it has more factors of 3 than X does.

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Re: X is the least common multiple of 96, 196, 300, which number below is  [#permalink]

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24 Nov 2019, 08:05
Bunuel wrote:
X is the least common multiple of 96, 196, 300, which number below is not factor of X?

A. 600
B. 700
C. 900
D. 2100
E. 4900

$$f(96):2^53$$
$$f(196):2^27^2$$
$$f(300):2^2*5^23$$
$$lcm(96,196,300):3*2^55^27^2$$

Ans (C): 9 is not factor of X
Re: X is the least common multiple of 96, 196, 300, which number below is   [#permalink] 24 Nov 2019, 08:05
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