If `f(x) and g(x)` are periodic functions with periods 7 and 11, respectively, then the period of `f(x)=f(x)g(x/5)-g(x)f(x/3)` is

Options

A. 177
B. 222
C. 433
D. 1155

Explanation

To find the period of the function \( f(x) = f(x)g\left(\frac{x}{5}\right) - g(x)f\left(\frac{x}{3}\right) \), where \( f(x) \) has a period of 7 and \( g(x) \) has a period of 11, we will follow these steps:

### Step 1: Determine the period of \( f\left(\frac{x}{3}\right) \)

The period of \( f(x) \) is 7. When we replace \( x \) with \( \frac{x}{3} \), the new period can be calculated as follows:

\[ \text{New Period} = \frac{\text{Original Period}}{\text{Coefficient of } x} = \frac{7}{\frac{1}{3}} = 7 \times 3 = 21 \]

### Step 2: Determine the period of \( g\left(\frac{x}{5}\right) \)

The period of \( g(x) \) is 11. When we replace \( x \) with \( \frac{x}{5} \), the new period can be calculated as follows:

\[ \text{New Period} = \frac{\text{Original Period}}{\text{Coefficient of } x} = \frac{11}{\frac{1}{5}} = 11 \times 5 = 55 \]

### Step 3: Determine the period of the first term \( f(x)g\left(\frac{x}{5}\right) \)

The period of the product \( f(x)g\left(\frac{x}{5}\right) \) is the least common multiple (LCM) of the periods of \( f(x) \) and \( g\left(\frac{x}{5}\right) \):

- Period of \( f(x) \) = 7
- Period of \( g\left(\frac{x}{5}\right) \) = 55

To find the LCM of 7 and 55, we note that 7 is a prime number and does not divide 55. Therefore:

\[ \text{LCM}(7, 55) = 7 \times 55 = 385 \]

### Step 4: Determine the period of the second term \( g(x)f\left(\frac{x}{3}\right) \)

The period of the product \( g(x)f\left(\frac{x}{3}\right) \) is the LCM of the periods of \( g(x) \) and \( f\left(\frac{x}{3}\right) \):

- Period of \( g(x) \) = 11
- Period of \( f\left(\frac{x}{3}\right) \) = 21

To find the LCM of 11 and 21, we note that 11 is a prime number and does not divide 21. Therefore:

\[ \text{LCM}(11, 21) = 11 \times 21 = 231 \]

### Step 5: Determine the overall period of \( f(x) \)

Now, we need to find the LCM of the two periods we calculated:

- Period of \( f(x)g\left(\frac{x}{5}\right) \) = 385
- Period of \( g(x)f\left(\frac{x}{3}\right) \) = 231

To find the LCM of 385 and 231, we can factor them:

- \( 385 = 5 \times 7 \times 11 \)
- \( 231 = 3 \times 7 \times 11 \)

The LCM is given by taking the highest power of each prime factor:

\[ \text{LCM}(385, 231) = 3^1 \times 5^1 \times 7^1 \times 11^1 = 3 \times 5 \times 7 \times 11 = 1155 \]

### Final Answer

Thus, the period of \( f(x) \) is \( 1155 \).

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