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@, $, and & are positive integers. If $ equals the square ro
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28 Jan 2020, 01:54
28 Jan 2020, 01:54
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@, $, and & are positive integers. If $ equals the square root of @, and if & equals the sum of @ and $, which of the following could be the value of &?
Indicate all such values.
[A] 21
[B] 30
[C] 45
[D] 72
[E] 100
[F] 331
Indicate all such values.
[A] 21
[B] 30
[C] 45
[D] 72
[E] 100
[F] 331
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Re: @, $, and & are positive integers. If $ equals the square ro
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28 Jan 2020, 03:51
28 Jan 2020, 03:51
1
huda wrote:
@, $, and & are positive integers. If $ equals the square root of @, and if & equals the sum of @ and $, which of the following could be the value of &?
Indicate all such values.
[A] 21
30
[C] 45
[D] 72
[E] 100
[F] 331
Indicate all such values.
[A] 21
30
[C] 45
[D] 72
[E] 100
[F] 331
\($ = \sqrt{@}\)
\(& = @ + $\)
\(=> & = @ + \sqrt{@}\)
Thus, \(&\) is equal to the sum of a number and that number's square-root
Since the numbers are positive integers, \(@\) must be a perfect square
Possible values of \(@\) are:
\(@\) = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
\(\sqrt{@}\) = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
\(&\) = 2, 6, 12, 20, 30, 42, 56, 72, 90, 110
We need to still verify Option F:
289 + \sqrt{289} = 306
324 + \sqrt{324} = 342
Thus, 331 cannot be expressed.
[b]An Alternate approach could have been:
\(=> & = @ + \sqrt{@}\)
\(=> & = \sqrt{@} * (\sqrt{@} + 1)\)
Since the numbers are positive integers, \(@\) must be a perfect square
Thus, \(&\) is the product of 2 consecutive integers. Only 30 = 5 * 6 and 72 = 8 * 9 satisfy
Thus, Options B and D are possible.
gmatclubot
Re: @, $, and & are positive integers. If $ equals the square ro [#permalink]
28 Jan 2020, 03:51
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