By what least number should the following number be divided to get a perfect square. a . 1275
The Correct Answer and Explanation is:
To determine the least number by which 1275 should be divided to obtain a perfect square, we begin by prime factorizing 1275.
Step 1: Prime Factorization of 1275
Let’s divide by the smallest prime numbers:
- 1275 ÷ 3 = 425
- 425 ÷ 5 = 85
- 85 ÷ 5 = 17
- 17 is a prime number
So,
1275 = 3 × 5 × 5 × 17
or
1275 = 3 × 5² × 17
Step 2: Understanding Perfect Squares
A perfect square has all prime factors in even powers.
From the factorization:
- 3 appears once → odd
- 5 appears twice → even
- 17 appears once → odd
To make all the exponents even, we need to eliminate the primes that appear an odd number of times. So we must remove both 3 and 17 from the product.
That means we should divide 1275 by 3 × 17 = 51.
Final Answer:
The least number by which 1275 should be divided to get a perfect square is 51.
Explanation:
To find the smallest number by which 1275 must be divided to produce a perfect square, we begin by factorizing it into its prime components. Prime factorization reveals the building blocks of a number and helps identify whether it can form a perfect square. A perfect square is a number whose prime factors all occur in even powers. For example, 36 = 2² × 3², which is a perfect square because each prime number is raised to an even power.
Starting with 1275, we divide it by the smallest primes:
- 1275 ÷ 3 = 425
- 425 ÷ 5 = 85
- 85 ÷ 5 = 17
This gives us the prime factorization: 1275 = 3 × 5² × 17.
Here, 3 and 17 are each raised to the power of one, which are odd powers. To make the number a perfect square, we must remove these two factors because they prevent the number from having only even exponents.
Removing 3 and 17 means dividing the number by 3 × 17 = 51. After this division:
1275 ÷ 51 = 25
Now 25 is a perfect square (since 25 = 5²), confirming our answer is correct. Therefore, dividing 1275 by 51 gives us the nearest perfect square with the smallest possible divisor.
So, the least number by which 1275 must be divided to get a perfect square is 51.
