If M is the least common multiple of 90, 196, and 300, which of the

IF M is the least common multiple of 90,196 and 300, which of the following is NOT a factor of M?

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

90 = 2 * 3 * 3 * 5
196 = 2 * 2 * 7 * 7
300 = 2 * 2 * 3 * 5 * 5
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LCM = 2 * 2 * 3 * 3 * 5 * 5 * 7 * 7
(TWO 2, TWO 3, TWO 5, TWO 7)

600 = 2 * 2 * 2 * 3 * 5 * 5
700 = 2 * 2 * 5 * 5 * 7
900 = 2 * 2 * 3 * 3 * 5 * 5
2100 = 2 * 2 * 3 * 5 * 5 * 7
4900 = 2 * 2 * 5 * 5 * 7 * 7

If M is the least common multiple of 90, 196, and 300, which of the following is NOT a factor of M?

Rule of Thumb: whenever you see Least Common Multiple (LCM), think of factors and preferebly prime factorization. Every non-prime integer can be factored into only prime numbers. So lets start prime factoring:
[P.S. if you need help in prime factoring tell us]
For easy start, whenever you see an even integer, start with the prime factor 2

90: 3 x 30 --> 3 x 3 x 10 --> 3 x 3 x 2 x 5
196: 2 x 98 --> 2 x 2 x 49 --> 2 x 2 x 7 x 7
300: 2 x 150 --> 2 x 2 x 75 --> 2 x 2 x 3 x 25 --> 2 x 2 x 3 x 5 x 5

LCM of the three number must contain all prime factors of each and every one of the three numbers : 90, 196, and 300

LCM : 2 x 2 x 3 x 3 x 5 x 5 x 7 x 7 = M

A factor of M must contain one or more, but limited to the available ones, of the prime factors of LCM [M]

A. 600 : 2 x 3 x 2 x 5 x 2 x 5 [ uses three 2's --> Not a Factor ]
B. 700 : 2 x 5 x 2 x 5 x 7 [ a factor of M ]
C. 900 : 3 x 3 x 2 x 2 x 5 x 5 [ a factor of M ]
D. 2,100 : 7 x 3 x 2 x 5 x 2 x 5 [ you tell me ]
E. 4,900 : 7 x 7 x 2 x 5 x 2 x 5 [ yes indeedy ]


It is A

90 = 3^2 x 2 x 5
196 = 7^2 x 2^2
300 = 3 x 5^2 x 2^2

so the least common = 7^2 x 2^2 x 3^2 x 5^2 = 7^2 x 3^2 x 100 = (7x3)^2 x 100

A. 6 = 3 x 2 <--not
B. 7 <-ok
C. 9 = 3^2 <- ok
D. 21 = 7 x 3 <- ok
E. 49 = 7^2 <-ok

the answer is A

The key to this problem is breaking down the numbers into its prime factors. After that its a piece of cake!

90 -> 3*3*5*2
196 -> 2*2*7*7
300 -> 3*2*2*5*5

M(LCM) = multiply all the factors (pick the highest power of the common factor)

M(LCM) = \(2^2*3^2*5^2*7^2\)

The question asks which is NOT A FACTOR of M:
Only A(600) which has an additional factor 2 is not a factor of M
600 -> \(2^3*3*5^2\) => it has an additional factor of 2 which is not present in M(only 2 factors of '2' is present here)

Here is @lyla4's answer in a manner understandable to late bloomers like me

LCM is the product of the greatest power of each prime that appears in any of the numbers.
90 = 3^2 x 2 x 5
196 = 7^2 x 2^2
300 = 3 x 5^2 x 2^2

So the least common multiple is = 7^2 x 2^2 x 3^2 x 5^2 = (7x3)^2 x 100
Divide each answer choice and the LCM by 100
Therefore LCM = (7x3)^2

Now we are looking for the choice in which, LCM/choice is not an integer.

A. 6 = 3 x 2 <--not
B. 7 <-ok
C. 9 = 3^2 <- ok
D. 21 = 7 x 3 <- ok
E. 49 = 7^2 <-ok

the answer is A

The least common multiple of 90, 196, and 300 = (2)(2)(3)(3)(5)(5)(7)(7)
So, M = (2)(2)(3)(3)(5)(5)(7)(7)

Now check the answer choices...
A) 600
600 = (2)(2)(2)(3)(5)(5)
For 600 to be a factor of M, there must be three 2's, one 3 and two 5's "hiding" in the prime factorization of M. Since, M only has two 2's in its prime factorization, 600 is NOT a factor of M.

Answer:

For more on the relationship between factor and prime factorization, watch this video: https://www.gmatprepnow.com/module/gmat ... /video/825

ASIDE: I thought it might be useful to show one way to find the LCM of large numbers:

Hi All,

This question is essentially about prime-factorization. Here's a simple example of that concept:

What is the least common multiple of 10 and 15. Now you probably already know that the LCM is 30, but here's WHY it's 30...

10 = (2)(5)
15 = (3)(5)

When looking for the LCM, we need to multiply all of the prime factors of the numbers involved. However, each instance of a prime that shows up in both numbers should be counted just once (here, there's one 5 in both numbers, so we count that as just ONE 5 and not two 5s). This gives us...

(2)(3)(5) = 30

We can then use those primes to figure out all of the divisors of the LCM:

1
2
3
5
(2)(3) = 6
(2)(5) = 10
(3)(5) = 15
(2)(3)(5) = 30

The exact same concept applies to this question - it's just that there's a lot more math work involved:

90 = (2)(3)(3)(5)
196 = (2)(2)(7)(7)
300 = (2)(2)(3)(5)(5)

The LCM of these three numbers will include two 2s, two 3s, two 5s and two 7s:

(2)(2)(3)(3)(5)(5)(7)(7)

At this point, you should NOT multiply those numbers together - we're just going keep them as a reference point so that we can find the one answer that is NOT a possible factor from that list:

Let's start with the easiest option first:

Answer A: 600 = (2)(2)(2)(3)(5)(5)

Notice how this number hast THREE 2s. This number is NOT possible given the list of primes that we have to work with, so it cannot be a factor of M.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Given that M is the least common multiple of 90,196, and 300 and we need to find which of the following is NOT a factor of M

Lets find the value of M first

90 = 2*3*3*5 = \(2^1 * 3^2 * 5^1\)
196 = 4*49 = \(2^2 * 7^2\)
300 = 2*2*3*5*5 = \(2^2 * 3^1 * 5^2\)

To find the LCM take all the terms and take their highest power in all of the above.

=> M = LCM(90, 196, 300) = \(2^2 * 3^2 * 5^2 * 7^2\) = 9 * 49 * 100
=> The Answer choices should divide the LCM

A. 600 Clearly, 600 is NOT a factor of the LCM as 600 = 6*100 and M = 9 * 49 * 100 and 9 * 49 is NOT DIVISIBLE by 6.
In exam we don't need to test further but I am solving to complete the solution.

B. 700 = 7 * 100. M = 441 * 100 and 9 * 49 = is divisible by 7 => POSSIBLE

C. 900 = 9 * 100. M = 441 * 100 and 9 * 49 is divisible by 9 => POSSIBLE

D. 2,100 = 21 * 100. M = 441 * 100 and 9 * 49 is divisible by 21 => POSSIBLE

E. 4,900 = 49 * 100. M = 441 * 100 and 9 * 49 is divisible by 49 => POSSIBLE

So, Answer will be A
Hope it helps!

To learn more about LCM and GCD watch the following videos




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First we can break 90, 196, and 300 to primes to determine the LCM of those numbers:

90 = 9 x 10 = 2^1 x 3^2 x 5^1

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

300 = 30 x 10 = 2^2 x 3^1 x 5^2

Thus, the LCM is 2^2 x 3^2 x 5^2 x 7^2

Next, we can break down our answer choices into primes:

600 = 60 x 10 = 6 x 10 x 10 = 2^3 x 3^1 x 5^2

Since 600 has three 2s, it cannot be a factor of M.

Answer: A
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I thought the question was about 2 number 90196 and 300. So there can be no space after the comma to separate different numbers? Or it is just a typo?

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Typo. Edited. Thank you!
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Asked: If M is the least common multiple of 90, 196, and 300, which of the following is NOT a factor of M?

90 = 2*3^2**5
196 = 2^2*7^2
300 = 2^2*3*5^2

M = LCM(90,196,300) = 2^2*3^2*5^2*7^2

A. 600 = 2^3*3*5^2 : NOT A FACTOR OF M
B. 700
C. 900
D. 2,100
E. 4,900

IMO A

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