Find 1938 is the perfect cube or not ?

Answer

To determine whether 1938 is a perfect cube, we will follow these steps:

### Step 1: Find the prime factorization of 1938
To start, we need to divide 1938 by the smallest prime number, which is 2.

- **1938 ÷ 2 = 969**

Next, we factor 969. Since 969 is odd, we try the next smallest prime number, which is 3.

- **969 ÷ 3 = 323**

Now we factor 323. The next prime number is 5, but it does not divide 323. We try the next prime number, which is 17.

- **323 ÷ 17 = 19**

Finally, 19 is a prime number itself.

So, the complete prime factorization of 1938 is:
- **1938 = 2 × 3 × 17 × 19**

### Step 2: Check the triplet condition
A number is a perfect cube if all the prime factors in its factorization are present in triplets (i.e., each prime factor's exponent must be a multiple of 3).

From the prime factorization:
- 2 appears once (1 time)
- 3 appears once (1 time)
- 17 appears once (1 time)
- 19 appears once (1 time)

### Step 3: Analyze the exponents
Since none of the prime factors (2, 3, 17, 19) are present in triplets (i.e., none of the exponents are multiples of 3), we conclude that:

### Conclusion
1938 is **not a perfect cube**.

---