Let `'**'` be a binary operation on N given by:
`a**b=LCM (a,b)` for all `a, b inN`.
Find `20**16`

Answer

To find \( 20 ** 16 \), where the operation \( ** \) is defined as the least common multiple (LCM) of two natural numbers, we will follow these steps:

### Step 1: Identify the numbers
We need to find \( 20 ** 16 \), which is defined as \( \text{LCM}(20, 16) \).

### Step 2: Prime Factorization
We will start by finding the prime factorization of both numbers:
- The prime factorization of \( 20 \) is:
\[ 20 = 2^2 \times 5^1 \]
- The prime factorization of \( 16 \) is:
\[ 16 = 2^4 \]

### Step 3: Determine the LCM
To find the LCM, we take the highest power of each prime factor that appears in the factorizations:
- For the prime \( 2 \), the highest power is \( 2^4 \) (from \( 16 \)).
- For the prime \( 5 \), the highest power is \( 5^1 \) (from \( 20 \)).

Thus, we can express the LCM as:
\[ \text{LCM}(20, 16) = 2^4 \times 5^1 \]

### Step 4: Calculate the LCM
Now we will calculate \( 2^4 \times 5^1 \):
\[ 2^4 = 16 \quad \text{and} \quad 5^1 = 5 \]
So,
\[ \text{LCM}(20, 16) = 16 \times 5 = 80 \]

### Conclusion
Therefore, \( 20 ** 16 = 80 \).

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