Find the least number that is multiplied to 1176 t0 make it perfect square.
Answer = 6
1176 = 2*2*7*7*(3*2)
1176 = 4 × 49 * (6)
In above eqn, 4 and 49, both are perfect squares, but 6 isn't. So if we multiply 1176 by 6 (i.e. 1176 * 6 = 7056) we get 7056, which is a square of 84.
I don't know if this is right or wrong but thanks a lot
To find the least number that must be multiplied by 1176 to make it a perfect square, we need to factorize 1176 and determine the missing factors.
The prime factorization of 1176 is:
1176 = 2 * 2 * 2 * 3 * 7 * 7
To make it a perfect square, we need each prime factor to have an even exponent.
The prime factorization of a perfect square should have all exponents as even numbers, so we need to find the missing factors:
1176 = 2^3 * 3 * 7^2
From the prime factorization, we can see that the missing factor to make it a perfect square is 2^1.
Therefore, the least number that must be multiplied by 1176 to make it a perfect square is 2.
If we multiply 1176 by 2, we get 2352, which is a perfect square (48^2).
To find the least number that, when multiplied by 1176, will make it a perfect square, we need to factorize 1176 and analyze the powers of the prime factors.
Step 1: Prime Factorization of 1176
To find the prime factorization of 1176, we can start by dividing it by two repeatedly until we can no longer divide it evenly:
1176 ÷ 2 = 588
588 ÷ 2 = 294
294 ÷ 2 = 147
147 is not divisible by 2, so let's try the next prime number, which is 3:
147 ÷ 3 = 49
Now, 49 is not divisible by 2 or 3, so let's try the next prime number, which is 7:
49 ÷ 7 = 7
Finally, we reached 7, which is a prime number. Therefore, the prime factorization of 1176 can be written as:
1176 = 2^3 × 3^1 × 7^2
Step 2: Determine the Powers
To make 1176 a perfect square, each prime factor needs to occur in pairs. We have:
2^3 × 3^1 × 7^2
In order to make the powers of 2 and 3 even, we need to multiply 1176 by 2^1 (since 2^3 already exists) and 3^1 (since 3^1 already exists). Also, to make the power of 7 even, we don't need to multiply by 7 since it's already even.
Step 3: Calculate the Least Number
Now we multiply the prime factors together:
2^1 × 3^1 × 7^2 = 2 × 3 × 49 = 294
Therefore, the least number that can be multiplied by 1176 to make it a perfect square is 294.