Let’s solve the given problems step-by-step:
Problem 9(c): Find the smallest number which must be subtracted from 6,085 to make it a perfect square.
To solve this problem, we need to determine the nearest perfect square that is less than or equal to 6,085 and then find the difference between 6,085 and this nearest perfect square.
1. Identify the integer part of the square root of 6,085.
[tex]\[
\sqrt{6,085} \approx 78
\][/tex]
2. The nearest perfect square less than or equal to 6,085 would be the square of 78.
[tex]\[
78^2 = 6,084
\][/tex]
3. Subtract this perfect square from 6,085 to find the smallest number that needs to be subtracted.
[tex]\[
6,085 - 6,084 = 1
\][/tex]
So, the smallest number that must be subtracted from 6,085 to make it a perfect square is 1.
---
Problem 10: Find the smallest number by which 1,48,176 must be divided to make it a perfect cube.
To solve this problem, follow these steps:
1. Factorize 1,48,176 into its prime factors.
2. Check the exponents of each prime factor to determine if the number is a perfect cube. For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.
3. Identify the prime factors whose exponents are not multiples of 3 and raise them to the required power to make the total exponent a multiple of 3.
4. The product of these prime factors raised to the required power gives the smallest number that needs to be divided into 1,48,176 to make it a perfect cube.
Given the factorization and analysis:
[tex]\[
\text{Smallest number} = 2
\][/tex]
Thus, the smallest number by which 1,48,176 must be divided to make it a perfect cube is 2.
