V.2 Find the smallest number by which 1100 must be multiplied so that the product becomes a perfect square. Also, find the square root of the perfect square so obtained.
The Correct Answer and Explanation is:
To solve the problem:
V.2 Find the smallest number by which 1100 must be multiplied so that the product becomes a perfect square. Also, find the square root of the perfect square so obtained.
✅ Correct Answer:
Smallest number = 11
Perfect square = 1100 × 11 = 12100
Square root = √12100 = 110
📘 Explanation (Approx. 300 words):
To make a number into a perfect square, we need to ensure that all the prime factors of the number appear in even powers. Let us begin by performing the prime factorization of 1100.
Step 1: Prime factorization of 11001100=11×100=11×(10×10)=11×(2×5)×(2×5)=11×22×521100 = 11 × 100 = 11 × (10 × 10) = 11 × (2 × 5) × (2 × 5) = 11 × 2^2 × 5^21100=11×100=11×(10×10)=11×(2×5)×(2×5)=11×22×52
So,1100=22×52×1111100 = 2^2 × 5^2 × 11^11100=22×52×111
Now, we observe the powers of the prime factors:
- 222^222: Even power – this is fine.
- 525^252: Even power – this is fine.
- 11111^1111: Odd power – this needs to be made even.
To make the power of 11 even, we must multiply the number by another 11. This will give:1100×11=(22×52×111)×11=22×52×1121100 × 11 = (2^2 × 5^2 × 11^1) × 11 = 2^2 × 5^2 × 11^21100×11=(22×52×111)×11=22×52×112
Now, all prime factors have even powers, so the product is a perfect square.
Step 2: Find the square root of the resulting perfect square
We now compute:1100×11=121001100 × 11 = 121001100×11=12100
Take the square root of 12100:12100=22×52×112=2×5×11=110\sqrt{12100} = \sqrt{2^2 × 5^2 × 11^2} = 2 × 5 × 11 = 11012100=22×52×112=2×5×11=110
🟩 Final Answer:
- Smallest number to multiply by = 11
- Perfect square = 12100
- Square root = 110
