Question

In the given question, there are two statements marked I and II. Decide which of the statements are sufficient to answer the question. Choose your answer from the given alternative.

What is the value of integer x ?
(I) The least common multiple of x and 45 is 225.
(II) The least common multiple of x and 20 is 300.

• A. Statement I alone is sufficient to answer the problem.
• B. Statement II alone is sufficient to answer the problem.
• C. Statement I and II both are needed.
• D. Statement I and II both are not sufficient.

Answer 1

Answer: (C)

Statement I and II both are needed.

(C): Try to determine the value of x using the LCM of x and certain other integers.

(1) INSUFFICIENT:

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

Notice on the chart to the right that the LCM of x and 45 contains two 3's. Because 45 contains two 3's, x can contain zero, one, or two 3's. The LCM of x and 45 contains two 5's. Because 45 contains only ONE 5, 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.)

Number 3 5

x ? $\times$ ?

45 $3^2$ $\times$ $5^1$

LCM $3^2$ $\times$ $5^2$

Therefore, x can be any of the following numbers:

$x=5\times 5=25$

$x=3\times 5\times 5=75$

$x=3\times 3\times 5\times 5=225$

(2) INSUFFICIENT:

Statement (2) tells you that x and $20(2\times 2\times 5)$ have an LCM of $300(=2\times 2\times 3\times 5\times 5=2^{2}2\times 3^1\times 5^2)$.

The LCM of x and 20 contains two 2's. Because 20 contains two 2's, x can contain zero, one, or two 2's. The LCM of x and 20 contains one 3. Because 20 contains no 3's, x must contain exactly one 3.

Number 2 3 5

x ? $\times$ ? $\times$ ?

20 $2^2$ - $\times$ $5^1$

LCM $2^2$ $\times$ $3^1$ $\times$ $5^2$

Further more, the LCM of x and 20 contains two 5's. Because 20 contains one 5, x must contain exactly two 5's.

Therefore, x can be any of the following numbers:

$x=3\times 5\times 5=75.$

$x=2\times 3\times 5\times 5=150.$

$x=2\times 2\times 3\times 5\times 5=300.$

(1) AND (2) 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. The correct answer is (C).