Question

By what number should 3675 be multiplied to get a perfect square in each case? Also, find the number whose square is the new number. Find the sum of those 2 numbers.

Answer 1

Hint: To find the number which we have to multiply with 3675 to get a perfect square, we have to find the factors of 3675 by the prime factorization method. so, the "Prime Factorization" is finding which prime numbers multiply together to make the original number. To find the prime factorization there are some steps to follow: Step 1: Start by dividing the number by the first prime number 2 and continue dividing by 2 until you get a decimal or remainder. Then divide by 3, 5, 7, etc. until the only numbers left are prime numbers. Step 2: Write the number as a product of prime numbers.

Complete step-by-step answer:
So, prime factors of 3675 are
\[3675{\text{ }} = {\text{ }}3{\text{ }} \times {\text{ }}5{\text{ }} \times {\text{ }}5{\text{ }} \times {\text{ }}7 \times {\text{ }}7\]
In the prime factorization of 3675
We see that 5 and 7 are in pair but 3 is not in pair. To get 3 in pairs, we have to multiply 3 on both sides. the number on L.H.S. so obtained after multiplied by 3 will be a perfect square.
So, 3 is the required number.
Now \[3675 \times 3{\text{ }} = {\text{ }}3 \times 3 \times 5 \times 5 \times 7 \times 7\]
11025 = 3×3×5×5×7×7\[11025{\text{ }} = {\text{ }}3 \times 3 \times 5 \times 5 \times 7 \times 7\]
Taking Square root on both sides:
\[\sqrt {11025} {\text{ }} = {\text{ }}\sqrt {3 \times 3 \times 5 \times 5 \times 7 \times 7} \]
\[ = {\text{ }}3 \times 5 \times 7{\text{ }} = {\text{ }}105\]
So, 105 is the number whose square is 11025.
Sum of the number that we find = 105+3 = 108.

Note: A perfect square is an integer that is a square of an integer. In other words, it is the product of the same integer itself. Prime factorization is also known as A factor tree method; Greatest Common Factor; Least Common Multiple.
Example – 9 is the square number. It is written as 3× 3.