What is the value of integer x?

Question

(1) The least common multiple of x and 45 is 225.
(2) The least common multiple of x and 20 is 300.

Kudos for a correct  solution .

NickPapagiorgio · Accepted Answer

To find the LCM of any 2 numbers, you multiply the prime factorization at the highest power of each number.

1. Prime factors of 225 are 5^2 and 3^2. Prime factors of 45 are 3^2 and 5^1. That means the prime factors of x have to have at least 5^2 in them and either 3^2, 3^1, or 3^0 (i.e., equals 1 and not considered a prime number). Our options for x are 25 (5^2), 75 (3^1 x 5^2) and 225 (3^2 x 5^2).  Insufficient.

2. Prime factors of 300 are 5^2, 2^2, and 3^1. Prime factors of 20 are 5^1 and 2^2. That means the prime factors of x have to have at least a 3^1 and 5^2 in them and either 2^1, 2^2, or 2^0 (i.e., equals 1 and not considered a prime number). Our options are 75 (5^2 x 3^1), 150 (5^2 x 2^1 x 3^1), and 300 (5^2 x 2^2 x 3^1).  Insufficient.

Both of the statements together satisfy the condition that x=75  Correct answer is C.

reto · Answer

Statement 1:
Prime Factors of 45: 3, 3, 5
Prime Factors of 225: 3, 3, 5, 5
Therefore X contributes at least a 5 in order to build the LCM. X can be 5 or 10 or any other number adding another 5 to the factors creating the LCM of 225. Insufficient.

Statement 2:
Prime Factors of 20: 2, 2, 5
Prime Factors of 300: 2, 2, 5, 5
Therefore X contributes at least a 5 in order to build the LCM. X can be 5 or 25 or any other number adding another 5 to the factors creating the LCM of 300. Insufficient.

Statement 1+2 together: X has factors 2, 2, 3, 3, 5, 5. Could have another 7 or 2 - it's not clear for me. Therefore...

Answer E.

jaxndaxt145 · Answer

I feel like if X was 5 then the LCM of X and 45 would be 45 not 225.

There is a rule to find the LCM that you use the max power of each prime factor. I am not sure the official language.

I believe statement 1) x could be 25 ,75, or 225; x needs to be 5^2 and 3^0,1, or 2
 statement 2) x could be 75, 150 or 300; x needs to be 5^2, 3^2, and 2^0,1,or 2

The only one number the same is 75.

IMO C

jimmy02 · Answer

Statement 1: for LCM to be 225 x should be multiple of 5 i .e 5, 10 ...etc.
Not suff
Statement 2: for LCM to be 300 x should be multiple of 15 i.e. 15,30...etc.
Not sufficient

taking both together x should be 75 to satisfy both.

Answer is C.

GMATinsight · Answer

Question : x = ?

Statement 1: The least common multiple of x and 45 is 225.

45 = 3^2 * 5
225 = 3^2 * 5^2

Which leads us to a conclusion that  x must have 5^2  but there are more than one possibility of x
e.g x = 3 * 5^2 
or x = 3^2 * 5^2

Hence,   NOT SUFFICIENT

Statement 2: The least common multiple of x and 20 is 300.

20 = 2^2 * 5
300 = 2^2 * 3* 5^2

i.e.  x must have 3*5^2  but there are more than one possibility of x
e.g x = 3 * 5^2 
or x = 2 * 3 * 5^2

Hence,   NOT SUFFICIENT

Combining the two statements 
x must have 3*5^2
and x must have 5^2
Statement 1 also confirms that x can't be an even Number else LCM would have been even
Statement 2 also confirms that x can't be a multiple of 3^2 else LCM would have been a multiple of 3^2

Hence, x must be 3*5^2
Hence,   SUFFICIENT

Answer: option  C

reto · Answer

Of course, youre right NickPapagiorgio

Thanks for that. I find it easier with a table: Unbenannt.jpg If you have 2 numbers X and 45, the LCM will always take the largest count of any of the prime numbers. Statement 1 tells us that X must have 5^2 as a factor and less than or equal two of 3s >> (3^2 ,3^1 or 3^0). So several options available. >>> Insufficient. Statement 2 tells us that X has again 5^2 in its prime factorisation. It tells us that exactly 3^1 is also part of the prime factors BUT we do not know how many 2s we should count. Therefore insufficient. Together there is only one match, Statement 1 has no 2s. Combined we know that the prime factors of X are 2^0, 3^1 and 5^2. Answer C. I revoke my previous post herewith. But no KUDOS earned for that.

chetan2u · Answer

1) statement 1 tells us that x is a multiple of 225/45 or an extra 5 other than the ones in 45 that is 3,3,5.. so it can be 25,75,225 insuff
2) statement 2 tells us that x is a multiple of 300/20 or an extra 15 other than the ones in 20 that is 2,2,5.. so it can be 75,150,225 insuff

combined only 75 fits in suff
ans C

Bunuel · Answer

MANHATTAN GMAT OFFICIAL SOLUTION:

Try to determine the value of x using the LCM of x and certain other integers.

Statement (1)  tells you that x and 45 (3*3 *5) have an LCM of 225 (=3*3*5*5 = 3^2*5^2).

Notice that the LCM of x and 45 contains two 3 s. 45 contains two 3 s, so x can contain zero, one, or two 3 s. The LCM of x and 45 contains two 5 s. 45 contains only ONE 5, so x must contain exactly two 5 s. (If x contained more 5 s, the LCM would contain more 5 s. If x contained fewer 5 s, the LCM would contain fewer 5 s.)

Therefore x can be any of the following numbers:
x = 5*5 = 25
x = 3*5*5 = 75
x = 3*3*5*5 = 225

NOT SUFFICIENT.

Statement (2)  tells you that x and 20 (2*2*5) have an LCM of 300 ( = 2*2*3*5*5 = 2^2*3^1*5^2).

The LCM of x and 20 contains two 2 s. 20 contains two 2 s, so x can contain zero, one, or two 2 s. The LCM of x and 20 contains one 3. 20 contains NO 3 s, so x must contain exactly one 3. Furthermore, the LCM of x and 20 contains two 5 s. 20 contains one 5, so x must contain exactly two 5's.

Therefore x can be any of the following numbers:
x = 3*5*5 = 75.
x = 2*3*5*5 = 150.
x = 2*2*3*5*5 = 300.
NOT SUFFICIENT.

Statement (1) tells you that x could be 25, 75, or 225. Statement (2) tells you that x could be 75, 150, or 300. The only number that satisfies both of these conditions is x = 75. Therefore, you know that x must be 75. SUFFICIENT. The correct answer is (C): BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer: C.

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What is the value of integer x?

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