To solve this problem, we need to determine the smallest number by which we must multiply each given number to make it a perfect square. Let's break down the steps:
### Part (a): Number 10368
1. Prime Factorization:
To find the smallest multiplier, we first need to perform the prime factorization of 10368.
2. Examine Exponents:
Once we factorize 10368, we need to inspect the exponents of the prime factors. If any prime factor has an odd exponent, we need to balance it by multiplying the number by that prime to make all exponents even (since each prime factor must be raised to an even power to form a perfect square).
3. Determine Multiplier:
By examining the exponents, identify which prime factor or factors need to be supplemented.
### Part (b): Number 8112
1. Prime Factorization:
We will perform a similar process for the number 8112 by first determining its prime factors.
2. Examine Exponents:
Like with 10368, we must check the exponents of its prime factors to ensure each exponent is even.
3. Determine Multiplier:
Any prime factor with an odd exponent will indicate the smallest multiplier needed to adjust the number to a perfect square.
### Results:
From the previous steps, the smallest number by which:
- 10368 must be multiplied to make it a perfect square is: 2
- 8112 must be multiplied to make it a perfect square is: 3
Therefore, the solution is:
For 10368: Multiply by 2.
For 8112: Multiply by 3.
