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805+ Level|   Exponents|   Sequences|      
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pichitrat
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If x is a positive integer and <z> is the greatest multiple of ten les Updated on: 17 Mar 2019, 04:05
Originally posted by pichitrat on 17 Mar 2019, 03:36.
Last edited by Bunuel on 17 Mar 2019, 04:05, edited 1 time in total.
Renamed the topic and edited the question.
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A
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95% (hard)

Question Stats:

42% (02:31) correct 58% (02:13) wrong based on 228 sessions

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If x is a positive integer and <z> is the greatest multiple of ten less than or equal to z, what is the value of x-<x>?

(1) <x^14> + x^15 has a units digit of 7
(2) <x^15> + x^16 has a units digit of 1

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I'm a little confuse when you say "<z> is the greatest multiple of ten...." from there I'm a little lost. Would be glad if somebody can help.

Thanks!
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A
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vishumangal
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If x is a positive integer and <z> is the greatest multiple of ten les 17 Mar 2019, 03:56
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<z> means for eg <5> will be 0 as 0 is the only multiple of 10 which is less than 5 but if it is <20> then value will be 20 as 20 is the multiple of 10 which is equal to the given value.

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If x is a positive integer and <z> is the greatest multiple of ten les 19 Mar 2019, 07:11
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Solution shall be as per below:




pichitrat
If x is a positive integer and <z> is the greatest multiple of ten less than or equal to z, what is the value of x-<x>?

(1) <x^14> + x^15 has a units digit of 7
(2) <x^15> + x^16 has a units digit of 1
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Attachment:
Solution.jpg
Solution.jpg [ 67.26 KiB | Viewed 4339 times ]
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If x is a positive integer and <z> is the greatest multiple of ten les Updated on: 03 Sep 2020, 15:06
Originally posted by exc4libur on 26 May 2020, 07:02.
Last edited by exc4libur on 03 Sep 2020, 15:06, edited 1 time in total.
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pichitrat
If x is a positive integer and <z> is the greatest multiple of ten less than or equal to z, what is the value of x-<x>?

(1) <x^14> + x^15 has a units digit of 7
(2) <x^15> + x^16 has a units digit of 1

x-<x> is asking for the units
ie. x,<x>=33,30; x,<x>=51,50

(1) sufic

cycles of 3: 3,9,27,81.. [3,9,7,1]
cycles of 7: 7,49,343,___1.. [7,9,3,1]
cycles of 9: 9,1.. [9,1]

units(x^15)=7, 15/{4,2} has rem {1,3}
if units x=3: 3^15 has units of 7 correct
if units x=7: 7^15 has units of 3 invalid
if units x=9: 9^15 has units of 9 invalid

units of x is 3

(2) insufic

cycles of 1: always [1]
cycles of 3: [3,9,7,1]
cycles of 7: [7,9,3,1]
cycles of 9: [9,1]

units(x^16)=1, 16/{4,2,1} has rem 0
if units x=1: 1^16 has units of 1 correct
if units x=3: 3^16 has units of 1 correct
if units x=7: 7^16 has units of 1 correct
if units x=9: 9^16 has units of 9 invalid

units x could be {1,3,7}

ans (A)
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If x is a positive integer and <z> is the greatest multiple of ten les 28 Oct 2020, 11:48
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This question is asking for the units digit of x; for example 116 = <110>.

This question tests cyclicity:

Cyclicity of 3: 3, 9, 7, 1
Cyclicity of 4: 4, 6
Cyclicity of 7: 7, 9, 3, 1

Lets take a look at the statements:

(1) <x^14> + x^15 has a units digit of 7

Only one number to the 15th power has a units digit of 7.

Cyclicity of 3: 3, 9, 7, 1

Statement 1 is sufficient.

(2) <x^15> + x^16 has a units digit of 1

Multiple numbers to the 16th power have a units digit of 1: 1, 3, 7. Not sufficient.

Answer is A.
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If x is a positive integer and <z> is the greatest multiple of ten les 11 Aug 2022, 07:01
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