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16 Sep 2014, 01:26
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If \(357^x*117^y=a\), where \(x\) and \(y\) are positive integers, what is the units digit of \(a\)?
(1) \(100 \lt y^2 \lt x^2 \lt 169\)
(2) \(x^2-y^2=23\)
(1) \(100 \lt y^2 \lt x^2 \lt 169\)
(2) \(x^2-y^2=23\)
16 Sep 2014, 01:26
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Official Solution:
If \(357^x*117^y=a\), where \(x\) and \(y\) are positive integers, what is the units digit of \(a\)?
(1) \(100 \lt y^2 \lt x^2 \lt 169\).
Since both \(x\) and \(y\) are positive integers, \(x^2\) and \(y^2\) are perfect squares. Now, there are only two perfect squares in the given range: \(121=11^2\) and \(144=12^2\), so \(y=11\) and \(x=12\). Sufficient. (As the cyclicity of the units digit of 7 in integer power is 4, therefore the units digit of \(7^{23}\) is the same as the units digit of \(7^3\), which is 3).
(2) \(x^2-y^2=23\).
From this equation, we have \((x-y)(x+y) = 23\). Since 23 is prime and both \(x\) and \(y\) are positive integers, it follows that \(x-y=1\) and \(x+y=23\). Solving for \(x\) and \(y\), we find \(y=11\) and \(x=12\). Sufficient.
Answer: D
If \(357^x*117^y=a\), where \(x\) and \(y\) are positive integers, what is the units digit of \(a\)?
(1) \(100 \lt y^2 \lt x^2 \lt 169\).
Since both \(x\) and \(y\) are positive integers, \(x^2\) and \(y^2\) are perfect squares. Now, there are only two perfect squares in the given range: \(121=11^2\) and \(144=12^2\), so \(y=11\) and \(x=12\). Sufficient. (As the cyclicity of the units digit of 7 in integer power is 4, therefore the units digit of \(7^{23}\) is the same as the units digit of \(7^3\), which is 3).
(2) \(x^2-y^2=23\).
From this equation, we have \((x-y)(x+y) = 23\). Since 23 is prime and both \(x\) and \(y\) are positive integers, it follows that \(x-y=1\) and \(x+y=23\). Solving for \(x\) and \(y\), we find \(y=11\) and \(x=12\). Sufficient.
Answer: D
General Discussion
Avigano
Joined: 12 Nov 2015
Last visit: 03 Jul 2017
Posts: 24
Own Kudos:
Given Kudos: 126
Location: Uruguay
Concentration: General Management
Schools: Goizueta '19 (A)
GMAT 1: 610 Q41 V32
GMAT 2: 620 Q45 V31
GMAT 3: 640 Q46 V32
GPA: 3.97
12 Mar 2016, 14:02
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Probably very basic question, anyway..
why is x-y in statement 2 equal 1?
why is x-y in statement 2 equal 1?
