To solve this problem, we need to find the smallest number that can divide 235445 and turn it into a perfect square. A perfect square is a number that can be expressed as the product of an integer with itself. A key property of perfect squares is that they have even powers for all their prime factors. Let's break down the steps needed to solve this problem:
### Step 1: Prime Factorize 235445
First, we need to factorize 235445 into its prime factors:
[tex]\[ 235445 = 5 \times 47 \times 1001 \][/tex]
Next, we factorize 1001 further:
[tex]\[ 1001 = 7 \times 11 \times 13 \][/tex]
So, the complete prime factorization of 235445 is:
[tex]\[ 235445 = 5^1 \times 47^1 \times 7^1 \times 11^1 \times 13^1 \][/tex]
### Step 2: Determine the Smallest Divisor
For 235445 to become a perfect square, each prime factor must have an even power. Currently, all prime factors occur only once (i.e., have a power of 1), which is odd. To make the number a perfect square, we must divide it by each of these prime factors to remove them, resulting in the smallest number to divide by:
[tex]\[ 5 \times 47 \times 7 \times 11 \times 13 = 235445 \][/tex]
So, the smallest number that can divide 235445 to make it a perfect square is:
[tex]\[ 235445 \][/tex]
### Step 3: Calculate the Resulting Perfect Square
When we divide 235445 by 235445, we get:
[tex]\[ \frac{235445}{235445} = 1 \][/tex]
Thus, the resulting perfect square number is 1.
### Step 4: Calculate the Square Root
The square root of the resulting perfect square number 1 is:
[tex]\[ \sqrt{1} = 1 \][/tex]
### Conclusion:
1. The smallest number by which 235445 must be divided to make it a perfect square is 235445.
2. The square root of the number obtained after the division is 1.
