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26 Oct 2022, 01:30
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
67% (01:32) correct
33%
(01:43)
wrong
based on 72
sessions
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Which of the following is a perfect square?
A. \(\frac{(44!)(45!)}{3}\)
B. \(\frac{(46!)(45!)}{3}\)
C. \(\frac{(46!)(47!)}{3}\)
D. \(\frac{(48!)(47!)}{3}\)
E. \(\frac{(48!)(49!)}{3}\)
A. \(\frac{(44!)(45!)}{3}\)
B. \(\frac{(46!)(45!)}{3}\)
C. \(\frac{(46!)(47!)}{3}\)
D. \(\frac{(48!)(47!)}{3}\)
E. \(\frac{(48!)(49!)}{3}\)
26 Oct 2022, 03:10
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For a number to be a perfect square the power the primes numbers that constitute that number should be even.
As 3 is in the denominator, let's start by checking the power of three first.
* Power of 3 in 44! = 17
* Power of 3 in 45! = 21
As 45 is divisible by 3, the next number is divisible is 48, so the power of three for 46! and 47! will also be 21.
* Power of 3 in 48! = 22
As 45 is divisible by 3, the next number is divisible is 51, so the power of three for 49! will also be 22.
Because three is in the denominator it will reduce the power of three in the numerator, and for the resultant number to be a perfect square, the power of three on one of the factorial should be odd (so that when reduced by 1, the power becomes even) and the power of other factorial should be even.
If we evaluate the options -
A - \(\frac{3^{17} * 3^{21}}{3}\)
Resultant is an odd power of three, hence eliminate
B - \(\frac{3^{21} * 3^{21}}{3}\)
Resultant is an odd power of three, hence eliminate
C - \(\frac{3^{21} * 3^{21}}{3}\)
Resultant is an odd power of three, hence eliminate
D - \(\frac{3^{21} * 3^{22}}{3}\)
Resultant is an odd even power of three, keep.
E - \(\frac{3^{22} * 3^{22}}{3}\)
Resultant is an odd power of three, hence eliminate
IMO D
As 3 is in the denominator, let's start by checking the power of three first.
* Power of 3 in 44! = 17
* Power of 3 in 45! = 21
As 45 is divisible by 3, the next number is divisible is 48, so the power of three for 46! and 47! will also be 21.
* Power of 3 in 48! = 22
As 45 is divisible by 3, the next number is divisible is 51, so the power of three for 49! will also be 22.
Because three is in the denominator it will reduce the power of three in the numerator, and for the resultant number to be a perfect square, the power of three on one of the factorial should be odd (so that when reduced by 1, the power becomes even) and the power of other factorial should be even.
If we evaluate the options -
A - \(\frac{3^{17} * 3^{21}}{3}\)
Resultant is an odd power of three, hence eliminate
B - \(\frac{3^{21} * 3^{21}}{3}\)
Resultant is an odd power of three, hence eliminate
C - \(\frac{3^{21} * 3^{21}}{3}\)
Resultant is an odd power of three, hence eliminate
D - \(\frac{3^{21} * 3^{22}}{3}\)
Resultant is an odd even power of three, keep.
E - \(\frac{3^{22} * 3^{22}}{3}\)
Resultant is an odd power of three, hence eliminate
IMO D
15 Feb 2024, 14:42
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Alternatively, we know that a perfect square number multiply by another perfect square number will result in a perfect square number.
(44)!(45)!/3 =
(44)!45*(44)!/3
45(44)!(44)!/3 = 15(44)!(44)!
(44)!(44)! is a perfect square, but15 is not a perfect square so we reject this option.
Similarly,
(46)!(45)!/3 = 46(45)!(45)!/3
46/3 is not an integer. So we reject this option
(46)!(47)!/3 = 47(46)!(46)!/3
47/3 is not an integer. So we reject this option.
(48)!(47)!/3 = 48(47)!(47)!/3
48/3 = 16
16*(47)!(47)!
16 is a perfect square, also (47)!(47)! their product is a perfect square
This option is correct
(48)!(49)!/3 = 49(48)!(48)!/3
But 49/3 is not an integer. So we reject this option.
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(44)!(45)!/3 =
(44)!45*(44)!/3
45(44)!(44)!/3 = 15(44)!(44)!
(44)!(44)! is a perfect square, but15 is not a perfect square so we reject this option.
Similarly,
(46)!(45)!/3 = 46(45)!(45)!/3
46/3 is not an integer. So we reject this option
(46)!(47)!/3 = 47(46)!(46)!/3
47/3 is not an integer. So we reject this option.
(48)!(47)!/3 = 48(47)!(47)!/3
48/3 = 16
16*(47)!(47)!
16 is a perfect square, also (47)!(47)! their product is a perfect square
This option is correct
(48)!(49)!/3 = 49(48)!(48)!/3
But 49/3 is not an integer. So we reject this option.
Posted from my mobile device
