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If x and y are positive integers greater than 1, is the unit digit of

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ChandlerBong
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If x and y are positive integers greater than 1, is the unit digit of [#permalink] New post   21 Mar 2023, 23:12
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If x and y are positive integers greater than 1, is the unit digit of x * y greater than 5?

(A) x is a factor of 85.

(B) The units digit of y\(^2\) and y\(^7\) is the same.
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If x and y are positive integers greater than 1, is the unit digit of [#permalink] New post   22 Mar 2023, 00:49
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ChandlerBong wrote:
If x and y are positive integers greater than 1, is the unit digit of x * y greater than 5?

(A) x is a factor of 85.

(B) The units digit of y\(^2\) and y\(^7\) is the same.


A tricky one!

Statement 1

(A) x is a factor of 85.

x can be 5, 17, or 85. So the unit digit of x can be 5 or 7.

However, we don't know the unit digit of y. Hence statement 1 alone is not sufficient.

Eliminate A and D.

Statement 2

(B) The units digit of y\(^2\) and y\(^7\) is the same.

y can end with 1, 5, 6, or 0.

However, we don't know the unit digit of x. Hence statement 2 alone is not sufficient.

Eliminate B.

Combined

x * y can be greater than 5, example when x = 17 and y ends in 1.

or

x * y can be less than or equal to 5, example when x = 5 and y ends in 5.

Hence, we are getting both Yes and No as the answer.

Option E
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Re: If x and y are positive integers greater than 1, is the unit digit of [#permalink] New post   22 Mar 2023, 23:08
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ChandlerBong wrote:
If x and y are positive integers greater than 1, is the unit digit of x * y greater than 5?
(A) x is a factor of 85.
(B) The units digit of y\(^2\) and y\(^7\) is the same.

Solution:
Pre Analysis:
  • x and y are positive integers greater than 1
  • If are asked if the unit digit of xy is greater than 5 or not
  • This is a YES-NO question

Statement 1: x is a factor of 85
  • According to this statement, value of x can be 5, 17 or 85
  • With no information on the unit digit of y, we cannot say if unit digit of xy is greater than 5 or not
  • Unit digit can be 5 (for example x = 5 and y = 11), less than 5 (for example x = 5 and y = 12) or more than 5 (for example x = 17 and y = 11)
  • Thus, statement 1 alone is not sufficient and we can eliminate options A and D

Statement 2: The units digit of y\(^2\) and y\(^7\) is the same
  • According to this statement, cyclicity of y is either 1 or 5
  • We know that no digit has cyclicity of 5
  • Thus unit digit of y is \((y)_u=1,5,6\) or \(0\)
  • With no information on the unit digit of x, we cannot say if unit digit of xy is greater than 5 or not
  • Thus, statement 2 alone is also not sufficient

Combining:
  • From statement 1, \((x)_u=5,7\)
  • From statement 1, \((y)_u=1,5,6,7\)
  • Still unit digit can be 5 or more

Hence the right answer is Option E
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Re: If x and y are positive integers greater than 1, is the unit digit of [#permalink]
22 Mar 2023, 23:08
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