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05 Feb 2021, 15:01
Kudos
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
39% (02:10) correct
61%
(02:22)
wrong
based on 186
sessions
History
Date
Time
Result
Not Attempted Yet
How many divisors of 21,600 are perfect squares?
A. 12
B. 11
C. 10
D. 9
E. 8
A. 12
B. 11
C. 10
D. 9
E. 8
How many divisors of 21600 are perfect squares
07 Feb 2021, 23:36
Kudos
Bookmarks
21600=2^5*3^3*5^2
we can split this as : (2^4*3^2*5^2) *3*2
Here we are interested in squares so take the possible even powers part:
(2^2)^2*(3^2)^1*(5^2)1
now we can get the factors for this : (2+1)*(1+1)*(1+1) =3*2*2 =12
we can split this as : (2^4*3^2*5^2) *3*2
Here we are interested in squares so take the possible even powers part:
(2^2)^2*(3^2)^1*(5^2)1
now we can get the factors for this : (2+1)*(1+1)*(1+1) =3*2*2 =12
How many divisors of 21,600 are perfect squares?
Solution: Let's do prime factorization of 21,600 i.e. \(21,600 = 2^5 * 3^3 * 5^2\)
For a number to be a square the exponent of the prime ALL prime factors should be an even number.
Thus, from the above prime factorization \(2^2 , 3^2, 2^4, 5^2\) are some of the perfect squares. The combinations of these 4 squares are also perfect squares:
1. \(1^1 = 1\)
2. \(2^2 = 4\)
3. \(3^2 = 9\)
4. \(2^4 = 16\)
5. \(5^2 = 25\)
6. \(2^2 * 3^2 = 36\)
7. \(2^2 * 5^2 = 100\)
8. \(3^2 * 2^4 = 144\)
9. \(3^2 * 5^2 = 225\)
10. \(2^4 * 5^2 = 400\)
11. \(2^2 * 3^2 * 5^2 = 900\)
12. \(3^2 * 2^4 * 5^2 = 3600\)
Answer should be 12 divisors.
Solution: Let's do prime factorization of 21,600 i.e. \(21,600 = 2^5 * 3^3 * 5^2\)
For a number to be a square the exponent of the prime ALL prime factors should be an even number.
Thus, from the above prime factorization \(2^2 , 3^2, 2^4, 5^2\) are some of the perfect squares. The combinations of these 4 squares are also perfect squares:
1. \(1^1 = 1\)
2. \(2^2 = 4\)
3. \(3^2 = 9\)
4. \(2^4 = 16\)
5. \(5^2 = 25\)
6. \(2^2 * 3^2 = 36\)
7. \(2^2 * 5^2 = 100\)
8. \(3^2 * 2^4 = 144\)
9. \(3^2 * 5^2 = 225\)
10. \(2^4 * 5^2 = 400\)
11. \(2^2 * 3^2 * 5^2 = 900\)
12. \(3^2 * 2^4 * 5^2 = 3600\)
Answer should be 12 divisors.
