LCM of `15^(10),20^(12)` and N is `60^(12)` , then find the possible value of N? (a) 825 (b) 920 (c) 325 (d) None of these

Options

A. 825
B. 920
C. 325
D. None

Answer

To solve the problem of finding the possible value of N given that the LCM of \(15^{10}\), \(20^{12}\), and \(N\) is \(60^{12}\), we can follow these steps:

### Step 1: Factor the numbers involved
First, we need to factor each of the numbers involved in the LCM:

- \(15 = 3 \times 5\)
- \(20 = 2^2 \times 5\)
- \(60 = 2^2 \times 3 \times 5\)

### Step 2: Write the powers for each number
Now, we can express each number with its prime factorization raised to the respective powers:

- \(15^{10} = (3 \times 5)^{10} = 3^{10} \times 5^{10}\)
- \(20^{12} = (2^2 \times 5)^{12} = 2^{24} \times 5^{12}\)
- \(N\) can be expressed as \(N = 2^a \times 3^b \times 5^c\) for some non-negative integers \(a\), \(b\), and \(c\).

### Step 3: Determine the LCM
The LCM of the numbers is determined by taking the highest power of each prime factor present in any of the numbers:

- For \(2\): The highest power is \(2^{24}\) from \(20^{12}\).
- For \(3\): The highest power is \(3^{10}\) from \(15^{10}\).
- For \(5\): The highest power is \(5^{12}\) from \(20^{12}\).

Thus, the LCM can be expressed as:
\[ \text{LCM} = 2^{24} \times 3^{10} \times 5^{12} \]

### Step 4: Set the LCM equal to \(60^{12}\)
Now we need to express \(60^{12}\) in terms of its prime factors:
\[ 60^{12} = (2^2 \times 3 \times 5)^{12} = 2^{24} \times 3^{12} \times 5^{12} \]

### Step 5: Compare the LCM with \(60^{12}\)
Since we know the LCM of \(15^{10}\), \(20^{12}\), and \(N\) is equal to \(60^{12}\), we can equate the powers of the prime factors:

1. For \(2\): \(24\) (from LCM) = \(24\) (from \(60^{12}\)) → No contribution from \(N\).
2. For \(3\): \(10 + b = 12\) → \(b = 2\).
3. For \(5\): \(10 + c = 12\) → \(c = 2\).

### Step 6: Determine \(N\)
Now we can substitute the values of \(a\), \(b\), and \(c\) into the expression for \(N\):
\[ N = 2^a \times 3^b \times 5^c = 2^0 \times 3^2 \times 5^2 = 1 \times 9 \times 25 = 225 \]

### Step 7: Check the options
Now we need to check which of the given options can be a possible value for \(N\):
- (a) 825 = \(3 \times 5^2 \times 11\) → Not valid.
- (b) 920 = \(2^3 \times 5 \times 23\) → Not valid.
- (c) 325 = \(5^2 \times 13\) → Not valid.
- (d) None of these → Correct.

### Final Answer
The possible value of \(N\) is **None of these**.

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