What is the least number multiplied to 200, so that number obtained becomes multiple of 60?

Options

A. a)5
B. b)15
C. c)3
D. d)8

Answer

To find the least number that must be multiplied to 200 so that the product is a multiple of 60, we can follow these steps:

### Step 1: Prime Factorization of 60
First, we need to find the prime factorization of 60.
- 60 can be broken down as follows:
- 60 = 2 × 30
- 30 = 2 × 15
- 15 = 3 × 5
- Therefore, the prime factorization of 60 is:
- \( 60 = 2^2 \times 3^1 \times 5^1 \)

### Step 2: Prime Factorization of 200
Next, we find the prime factorization of 200.
- 200 can be broken down as follows:
- 200 = 2 × 100
- 100 = 2 × 50
- 50 = 2 × 25
- 25 = 5 × 5
- Therefore, the prime factorization of 200 is:
- \( 200 = 2^3 \times 5^2 \)

### Step 3: Compare the Prime Factorizations
Now, we compare the prime factorizations of 60 and 200:
- For 60: \( 2^2 \times 3^1 \times 5^1 \)
- For 200: \( 2^3 \times 5^2 \)

### Step 4: Identify Missing Factors
To make 200 a multiple of 60, we need to ensure that it has at least the same number of each prime factor as in 60:
- The factor of \( 2 \): 200 has \( 2^3 \) which is sufficient (since \( 2^3 \) ≥ \( 2^2 \)).
- The factor of \( 3 \): 200 has no factor of 3, so we need to multiply by \( 3^1 \).
- The factor of \( 5 \): 200 has \( 5^2 \) which is sufficient (since \( 5^2 \) ≥ \( 5^1 \)).

### Step 5: Calculate the Least Number to Multiply
Since the only missing factor is \( 3^1 \), we need to multiply 200 by 3 to make it a multiple of 60.

### Conclusion
The least number that must be multiplied to 200 so that the product is a multiple of 60 is **3**.

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