Finding Perfect Squares of n Factored as a^4*b^3*c^7

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In summary, when a number n is factorised into the form a^4*b^3*c^7, there are a total of 24 possible perfect squares that are factors of n. This is because for each 3 powers of a, there are two possible powers of b and for each of b there are 4 possibilities of c when each square is taken as the product of powers of each primes. However, during the combination of 2, 1, and 3, we won't get 1. The answer of 3*2*4 is obtained by multiplying the number of possible powers of a, b, and c.
  • #1
shravan
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a number n when factorised can be written as a^4*b^3*c^7.find number of perfect square which are factors of n.a,b,c are prime >2.
I have no idea how to start? please help.
 
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  • #2
If a, b and c are distinct primes then numbers like [itex]a^2[/itex] are perfect squares and factors of n. There are 2 possible perfect squares arising from a, only 1 from b but 3 possibilities from c so the question is how many combinations are possible with those perfect squares?
 
  • #3
Tide, how dare you miss such easy things? 3 possible perfect squares from a, 2 possible perfect squares from b and 4 possiblities from c of which in total three are same. You have missed the 0th power. Now the answer is 3*2*4. However during the combination of 2, 1 and 3 we won't get 1. Now the answer 3*2*4 is because for each 3 powers of a, there are two possible powers of b and for each of b there are 4 possibilities of c when each square is taken as the product of powers of each primes.
 
  • #4
vaishakh,

No, I didn't miss the zero power. I had to leave something for the student to think about!
 

1. What is a perfect square?

A perfect square is a number that is the product of two equal integers. For example, 9 is a perfect square because it is the product of 3 and 3.

2. How do you find the perfect square of n factored as a^4*b^3*c^7?

To find the perfect square of n factored as a^4*b^3*c^7, you need to first identify the largest possible square numbers that can be formed from the exponents of a, b, and c. In this case, the largest square numbers are 4, 3, and 7. We then take the square root of each of these numbers and multiply them together, giving us the perfect square of a^2*b*c^3.

3. Can you provide an example of finding the perfect square of n factored as a^4*b^3*c^7?

Sure, let's take the expression 16a^4*b^3*c^7. The largest square numbers that can be formed from the exponents are 4, 3, and 7. Taking the square root of each of these numbers gives us 2, 1.73, and 2.65. Multiplying these together gives us the perfect square of 16a^2*b*c^3.

4. Why is it important to find the perfect square of n factored as a^4*b^3*c^7?

Finding the perfect square of n factored as a^4*b^3*c^7 can help simplify expressions and make them easier to work with. It can also help in solving equations and identifying patterns in mathematical problems.

5. Can the same method be used for finding perfect squares with different exponents?

Yes, the same method can be used for finding perfect squares with different exponents. However, the square root of the largest possible square numbers will change depending on the exponents given.

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