Find the least number by which 1500 is to be multiplied so that the product becomes a perfect square Find the square root of the number obtained
The Correct Answer and Explanation is:
To solve this, we must find the least number by which 1500 should be multiplied so that the product is a perfect square, and then find the square root of the resulting perfect square.
Step 1: Prime Factorization of 1500
We begin by breaking 1500 into its prime factors:1500=15×100=(3×5)×(10×10)=3×5×(2×5)21500 = 15 \times 100 = (3 \times 5) \times (10 \times 10) = 3 \times 5 \times (2 \times 5)^21500=15×100=(3×5)×(10×10)=3×5×(2×5)2
Now let’s write this more clearly:1500=22×3×531500 = 2^2 \times 3 \times 5^31500=22×3×53
Step 2: Conditions for a Perfect Square
A perfect square has even powers of all its prime factors.
In the factorization:
- 222^222 is fine since 2 is raised to an even power.
- 313^131 is not a perfect square factor — the power is odd.
- 535^353 is also not a perfect square factor — the power is odd.
To make all powers even:
- Multiply by 3 to make 313^131 become 323^232
- Multiply by 5 to make 535^353 become 545^454
So, the least number to multiply by is:3×5=153 \times 5 = 153×5=15
Step 3: Multiply and Find the Product
1500×15=225001500 \times 15 = 225001500×15=22500
Step 4: Find the Square Root of the Result
Let’s find:22500\sqrt{22500}22500
We simplify:22500=225×100=225×100=15×10=150\sqrt{22500} = \sqrt{225 \times 100} = \sqrt{225} \times \sqrt{100} = 15 \times 10 = 15022500=225×100=225×100=15×10=150
Final Answer:
- Least number to multiply: 15
- Perfect square obtained: 22500
- Square root: 150
Explanation
To determine the least number by which 1500 must be multiplied to become a perfect square, we use the principle of prime factorization. A perfect square has even powers of all its prime factors. We start by factorizing 1500 into prime components. We find that 1500 equals 22×3×532^2 \times 3 \times 5^322×3×53. Here, the exponent of 2 is even, but the exponents of 3 and 5 are odd. For the product to be a perfect square, all prime exponents must be even.
To achieve this, we must supply enough of each prime to raise the exponents to even numbers. For the 3, we need one more factor of 3 to make it 323^232. For the 5, which is currently 535^353, we need another factor of 5 to raise it to 545^454. Therefore, we need to multiply 1500 by 3×5=153 \times 5 = 153×5=15.
Multiplying 1500 by 15 gives us 22500. This number has the prime factorization 22×32×542^2 \times 3^2 \times 5^422×32×54, where all exponents are now even. This confirms that 22500 is indeed a perfect square.
To find the square root, we take the square root of 22500. Since 22500 is 225×100225 \times 100225×100, we take the square root of each: the square root of 225 is 15 and the square root of 100 is 10. So the final square root is 15×10=15015 \times 10 = 15015×10=150.
This problem demonstrates how prime factorization can be used to systematically determine the smallest multiplier needed to create a perfect square and how to efficiently compute the square root of that result.
