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# If a and b are non-zero single digit integers such that a^b = 1, what

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Joined: 02 Sep 2009
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If a and b are non-zero single digit integers such that a^b = 1, what  [#permalink]

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07 Nov 2019, 03:17
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95% (hard)

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15% (03:01) correct 85% (02:39) wrong based on 47 sessions

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If a and b are non-zero single digit integers such that $$a^b = 1$$, what is the value of b?

(1) $$−1 ≤ b^a ≤ 1$$
(2) $$10a + b$$ raised to any positive power always has its units digit as b.

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Re: If a and b are non-zero single digit integers such that a^b = 1, what  [#permalink]

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07 Nov 2019, 11:57
1
from the given,
if a= 1, then b = any integer from -9 to 9 except 0
if a = -1, then b = any even integer from -8 to 8 except 0

from statement (1),
if a =1, then b = 1 or -1
if a = -1, then b = any integer from -9 to 9 except 0 --> insufficient

from statement (2),
if a =1, then b = 1,5,6 (mono-cyclic integers) or -5 ( which leads to a mono-cyclic integer 5)
if a =-1, then b = 5 or -5 (which lead to a number with 5 as its unit digit) --> insufficient

upon combining,
if a = 1, then b = 1 --> sufficient C
if a = -1, then b has no valid common value (which means that a is not equal -1)
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Re: If a and b are non-zero single digit integers such that a^b = 1, what  [#permalink]

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10 Nov 2019, 20:14
If a and b are non-zero single digit integers such that a^b=1, what is the value of b?

(1) −1≤b^a≤1
(2) 10a+b raised to any positive power always has its units digit as b.

Analysis of question stem tells us that a =1 since a is non zero integer and a^b =1.
As per option (1) b can be either -1 or 1 - hence not sufficient
As per option (2) values of b can be either 1,5 or 6 as 10a+b will always have unit digit of b and any power of b has to result in same unit digit as b. so only 3 values satisfy that. - not sufficient

Option (1) + (2) - sufficient as only number which satisfies both is 1. - hence Answer C
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Re: If a and b are non-zero single digit integers such that a^b = 1, what  [#permalink]

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18 Nov 2019, 12:03
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Re: If a and b are non-zero single digit integers such that a^b = 1, what  [#permalink]

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26 Nov 2019, 05:52
@kitipriyanka,i am no expert but I could help

If a and b are non-zero single digit integers such that a^b=1 what is the value of b?

(1) −1≤b^a≤1
(2) 10a+b10a+b raised to any positive power always has its units digit as b.
Before we answer the question , what non zero integers will give us a^b =1.
Either 1 raised to the power any integer or -1 raised to an even integer

Statement 1 : a=1,b=1 works for us.a=-1 and b=6 works as well.Insufficient

statement 2 : Now we are told in the stem that, a and b are non zero digits. Meaning 10a+b will have its units digit as b.We know that numbers with the base of 0,1,5 or 6 when raised to a positive power will retain the units digit of the base.Obviously,b can not be 0 but it can be 1 or 5 or 6.In sufficient

together if b=6 and a =-1 ,we satisfy both statements and the question stem.If b=1 and a=1,we satisfy both statements and the question stem as well.In both cases b has 2 different values of 1 and 6.Insufficient

Re: If a and b are non-zero single digit integers such that a^b = 1, what   [#permalink] 26 Nov 2019, 05:52
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