Question

Two numbers 12 and t are two positive numbers with some similar properties. What is the value of t.
1. The Least Common Multiple of the two numbers is 48.
2. The Greatest Common multiple of the two numbers is 4.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question ask.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hint:

Remember the fundamental property connecting two numbers with their LCM and GCD: \( \text{Product of Numbers} = \text{Product of LCM and GCD} \). This is a powerful tool for this type of question.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks for the specific value of a positive number \( t \), given that the other number is 12. The statements provide information about the Least Common Multiple (LCM) and Greatest Common Divisor (GCD) of 12 and t. Note: "Greatest Common multiple" in statement 2 is a common typo for "Greatest Common Divisor" (GCD).
Step 2: Key Formula or Approach:
For any two positive integers \( a \) and \( b \), there is a fundamental relationship between the numbers, their LCM, and their GCD:
\[ a \times b = \text{LCM}(a, b) \times \text{GCD}(a, b) \] In our case, \( a=12 \) and \( b=t \). So, \( 12 \times t = \text{LCM}(12, t) \times \text{GCD}(12, t) \).
Step 3: Detailed Explanation:
Analyze Statement (1): "The Least Common Multiple of the two numbers is 48."
We have LCM(12, t) = 48.
Let's use prime factorization. \( 12 = 2^2 \times 3^1 \). \( 48 = 2^4 \times 3^1 \).
The LCM is formed by taking the highest power of each prime factor present in either number.
Let \( t = 2^a \times 3^b \times \dots \).
LCM( \( 2^2 \times 3^1, 2^a \times 3^b \) ) = \( 2^{\max(2,a)} \times 3^{\max(1,b)} \).
This must equal \( 2^4 \times 3^1 \).
So, \( \max(2, a) = 4 \implies a = 4 \).
And \( \max(1, b) = 1 \implies b \) can be 0 or 1.
Possible values for t are: - If b=0, \( t = 2^4 = 16 \). (Check: LCM(12, 16) = 48. Correct.) - If b=1, \( t = 2^4 \times 3^1 = 48 \). (Check: LCM(12, 48) = 48. Correct.) Since t could be 16 or 48, Statement (1) is not sufficient.
Analyze Statement (2): "The Greatest Common Divisor of the two numbers is 4."
We have GCD(12, t) = 4.
This means t is a multiple of 4. So \( t = 4k \) for some integer k.
Also, t cannot be a multiple of 12 (otherwise the GCD would be 12). The common factors of 12 and t can't include 3.
Possible values for t: - If t = 4, GCD(12, 4) = 4. Possible. - If t = 8, GCD(12, 8) = 4. Possible. - If t = 16, GCD(12, 16) = 4. Possible. - If t = 20, GCD(12, 20) = 4. Possible. There are many possible values for t. Statement (2) is not sufficient.
Analyze Statements (1) and (2) Together:
From (1): LCM(12, t) = 48.
From (2): GCD(12, t) = 4.
Using the formula \( a \times b = \text{LCM}(a, b) \times \text{GCD}(a, b) \):
\[ 12 \times t = 48 \times 4 \] \[ 12t = 192 \] \[ t = \frac{192}{12} \] \[ t = 16 \] This gives a single, unique value for t. Therefore, the statements together are sufficient.
Step 4: Final Answer:
Neither statement alone is sufficient, but combining them provides a unique answer. The correct option is (C).