Answer :
To determine the smallest number by which 2475 must be multiplied to get a perfect square, let's go through the factorization and observe the conditions for forming a perfect square.
### Step-by-Step Solution:
1. Factorize the Number 2475:
Begin by finding the prime factorization of 2475. The factorization process involves dividing the number by the smallest prime number and continuing the division with the quotient until it is no longer divisible by that prime number. Repeat this process with subsequent prime numbers until you are left with 1.
[tex]\[ 2475 \div 5 = 495 \][/tex]
[tex]\[ 495 \div 5 = 99 \][/tex]
[tex]\[ 99 \div 3 = 33 \][/tex]
[tex]\[ 33 \div 3 = 11 \][/tex]
[tex]\[ 11 \div 11 = 1 \][/tex]
Thus, the prime factorization of 2475 is:
[tex]\[ 2475 = 5^2 \times 3^2 \times 11 \][/tex]
2. Analyzing the Exponents:
For a number to be a perfect square, every prime factor in its factorization must appear to an even power. Let's examine the exponents in our factorization:
- [tex]\(5^2\)[/tex] (Exponent is 2, which is even)
- [tex]\(3^2\)[/tex] (Exponent is 2, which is even)
- [tex]\(11^1\)[/tex] (Exponent is 1, which is odd)
Here, 11 appears only once, which is an odd exponent. To make it even, we need to multiply by another 11.
3. Forming a Perfect Square:
By multiplying 2475 by 11, we ensure that all prime factors in the resulting number have even exponents:
[tex]\[ 2475 \times 11 = (5^2 \times 3^2 \times 11^1) \times 11 = 5^2 \times 3^2 \times 11^2 \][/tex]
Now, each prime factor has an even exponent and thus the resulting number is a perfect square.
4. Conclusion:
The smallest number by which 2475 must be multiplied to obtain a perfect square is [tex]\( 11 \)[/tex].
So, the answer is [tex]\( \boxed{11} \)[/tex].
### Step-by-Step Solution:
1. Factorize the Number 2475:
Begin by finding the prime factorization of 2475. The factorization process involves dividing the number by the smallest prime number and continuing the division with the quotient until it is no longer divisible by that prime number. Repeat this process with subsequent prime numbers until you are left with 1.
[tex]\[ 2475 \div 5 = 495 \][/tex]
[tex]\[ 495 \div 5 = 99 \][/tex]
[tex]\[ 99 \div 3 = 33 \][/tex]
[tex]\[ 33 \div 3 = 11 \][/tex]
[tex]\[ 11 \div 11 = 1 \][/tex]
Thus, the prime factorization of 2475 is:
[tex]\[ 2475 = 5^2 \times 3^2 \times 11 \][/tex]
2. Analyzing the Exponents:
For a number to be a perfect square, every prime factor in its factorization must appear to an even power. Let's examine the exponents in our factorization:
- [tex]\(5^2\)[/tex] (Exponent is 2, which is even)
- [tex]\(3^2\)[/tex] (Exponent is 2, which is even)
- [tex]\(11^1\)[/tex] (Exponent is 1, which is odd)
Here, 11 appears only once, which is an odd exponent. To make it even, we need to multiply by another 11.
3. Forming a Perfect Square:
By multiplying 2475 by 11, we ensure that all prime factors in the resulting number have even exponents:
[tex]\[ 2475 \times 11 = (5^2 \times 3^2 \times 11^1) \times 11 = 5^2 \times 3^2 \times 11^2 \][/tex]
Now, each prime factor has an even exponent and thus the resulting number is a perfect square.
4. Conclusion:
The smallest number by which 2475 must be multiplied to obtain a perfect square is [tex]\( 11 \)[/tex].
So, the answer is [tex]\( \boxed{11} \)[/tex].
