What least number should be multiplied to 245 to make it a perfect square ?

Options

A. 2
B. 3
C. 5
D. 7

Answer

To determine the least number that should be multiplied to 245 to make it a perfect square, we can follow these steps:

### Step 1: Factor 245 into its prime factors.
To find the prime factors of 245, we can divide it by the smallest prime numbers:

- 245 is divisible by 5 (since it ends in 5):
\[ 245 \div 5 = 49 \]
- Next, we factor 49, which is \(7 \times 7\) or \(7^2\).

Thus, the prime factorization of 245 is:
\[ 245 = 5^1 \times 7^2 \]

### Step 2: Identify the powers of the prime factors.
In the prime factorization \(5^1 \times 7^2\):
- The power of 5 is 1 (which is odd).
- The power of 7 is 2 (which is even).

### Step 3: Determine what is needed for each prime factor to become a perfect square.
For a number to be a perfect square, all the powers in its prime factorization must be even.

- The power of 5 is odd (1), so we need one more 5 to make it even (2).
- The power of 7 is already even (2), so we do not need to add anything for 7.

### Step 4: Multiply the necessary factors.
To make 245 a perfect square, we need to multiply it by 5 (to make the power of 5 even):
\[ \text{Least number to multiply} = 5 \]

### Conclusion:
Thus, the least number that should be multiplied to 245 to make it a perfect square is **5**.

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