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23 Jun 2019, 07:03
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D
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Difficulty:
95%
(hard)
Question Stats:
35% (02:02) correct
65%
(01:49)
wrong
based on 79
sessions
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If x and y are positive integers and x*y = prime number greater than 2, what is the units’ digit of \(7^x + 9^y\)?
(1) 21 < y < 38
(2) x = 1
www.GMATinsight.com
(1) 21 < y < 38
(2) x = 1
www.GMATinsight.com
23 Jun 2019, 07:22
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Let the prime number be 'p.' Since, p is greater than 2, it's an odd prime.
Since a prime number has only 2 factors, 1 and the number itself, either (x,y) = (1,p) or (p,1)
Since we are asked about unit digit of the expression 7^x + 9^y, we will examine unit digit of the components of the expression individually:
We know that \(7^x\) ends in 7,9,3,1,7,9,3,1,7...
Similarly, \(9^y\) ends in 9,1,9,1,9...
Since x and y can only be odd numbers, \(7^x\) can only end in 7 or 3 and \(9^y\) can only end in 9.
So, the unit digit of the expression \(7^x + 9^y\) can either be 6 (unit digit of 7+9) or 2 (unit digit of 3+9).
Statement (1): y is some number which is not 1. So, x is 1. Hence, \(7^x\) has 7 as the unit digit. Thus, \(7^x+ 9^y\) will have 6 (unit digit of 7+9) as the unit digit. Sufficient.
Statement (1): Given that x is 1, implying \(7^x+ 9^y\) will have 6 as the unit digit (same logic as in statement (1). Sufficient.
Hence, D is the answer. Both statements are individually sufficient.
Since a prime number has only 2 factors, 1 and the number itself, either (x,y) = (1,p) or (p,1)
Since we are asked about unit digit of the expression 7^x + 9^y, we will examine unit digit of the components of the expression individually:
We know that \(7^x\) ends in 7,9,3,1,7,9,3,1,7...
Similarly, \(9^y\) ends in 9,1,9,1,9...
Since x and y can only be odd numbers, \(7^x\) can only end in 7 or 3 and \(9^y\) can only end in 9.
So, the unit digit of the expression \(7^x + 9^y\) can either be 6 (unit digit of 7+9) or 2 (unit digit of 3+9).
Statement (1): y is some number which is not 1. So, x is 1. Hence, \(7^x\) has 7 as the unit digit. Thus, \(7^x+ 9^y\) will have 6 (unit digit of 7+9) as the unit digit. Sufficient.
Statement (1): Given that x is 1, implying \(7^x+ 9^y\) will have 6 as the unit digit (same logic as in statement (1). Sufficient.
Hence, D is the answer. Both statements are individually sufficient.
Archit3110
Major Poster
23 Jun 2019, 10:04
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GMATinsight
given
x*y prime no >2 ; so either of x or y is 1 and other no would be odd integer
#1
21<y<38
only 2 is even prime integer rest all are odd so any no to the power of 9^y = will have units digit as 9
and x has to be 1 ; so 7^1+9^odd ; 7+9; 16 ; 6 is unit digit
#2
x=1
so y has to be any odd integer ; same logic as #1
unit digit = 6
IMO D
