The exponent of a^x is a multiple of 20, what is the minimum p

Question

The exponent of  a^{x}  is a multiple of 20, what is the minimum possible value of (a + x), if  a  is a factor of  x ? ( a & x are positive integers )

A. 10

B. 15

C. 20

D. 40

E. 50

foryearss · Answer

it is not an easy question : let me try : 
a ^x is a factor of 20 
a^x = 2^2 . 5 .y where y is an integer
a is a factor of x 
so x=a.k where k is an integer
a^a.k = 2^2 . 5 y 
now a cannot be <10 because we have 2s and a 5 in the right side 
a should at least be 10 
10^10*1 =10^10 is the least possible value for a^x (a=10 , x=10 )
so the least possible value for (x+a) is 20 
Answer : C

I would say that the level of this question is >700 
What do you think ?

generis · Answer

foryearss , I wonder about this part:
You wrote 
The prompt says: The exponent of  a^{x}   is a multiple of 20

x  must be a positive integer. The first positive multiple of 20 is 20.

How can  x=10 ?

foryearss · Answer

the whole exponent is a multiple of 20 , not x 
and i put
 a^x = 20 . y (20 is a factor of a^x ) (and y is what ever the rest of a^x could be )
a^x = 2^2 . 5 . y
now i want to start trying values for a 
can a be 2 for example ? no 
why ? because 2^(what ever ) will produce a number that only has 2s as factors . I want to produce a number that has both 2s and at least one 5 as factors .
can a be 3 ? no for the same reason , also a cannot be 4,5,6,7,8,9
now we reach 10 . 
a can be 10 , yes we can produce a number that has both 2s and 5 when raised to a power .
but what power of 10 ? we are interested in the least possible value .
if a=10 
and x=a.k ( a is a factor of x ) 
then x is at least 10 
so a^x =10^10
and that is a multiple of 20

chetan2u · Answer

exponent clearly means the power..
if it meant  a^x , you did not require to mention exponent, it would have been simply
a^x is a multiple of 20..

so if i take x as 20, the lowest factor of 20 will be 1..
so  a^x=1^{20} , +x=1+20=21..

foryearss · Answer

Oh , i have misunderstood the question ...

but why not saying x is a multiple of 20 ..

generis · Answer

... which is the answer I got.

Thanks,  chetan2u

foryearss  - at least you gave it a shot.

Maybe I misunderstand your statement, but
. . .
the prompt DOES say that  x , which = the  exponent  of  a^{x} ,  is a multiple of 20

foryearss · Answer

i meant why not saying : 
x is a multiple of 20 
instead of : 
the exponent of a^x is a multiple of 20 
Anyway , so now what is the answer ? 21 is not an option

generis · Answer

The question is flawed or the answers are wrong.

You ask why the question does NOT say "x is a multiple of 20"
instead of
"the exponent of a^x is a multiple of 20."

I did not write the question. 
I have no idea why the author wrote the question as s/he did.

GMATSkilled  - what is the OE?

CAMANISHPARMAR · Answer

10^{10}  making it 10 + 10 = 20. Therefore option C is the correct answer.

garvitbh11 · Answer

How is it possible ?? As question stem just said that Exponent which is "x" is a multiple of 20. 10 is not a multiple of 20. Please explain.

vishalkazone · Answer

Given:
i) a^x is multiple of 20: Means unit digit must be Zero.
ii) a is a factor of x means a< x or a= x, a cant be greater than x
iii) both a and x is positive integer.

we need to find minimum value of (a+x). look at the options with above results:
A: 10 no value of a and x can result to unit digit zero. ( multiple of 20) eliminate
B: 15 only 10^5 (a^x) can give unit digit zero but , condition 2 didn’t match a has to be less than or equal to x. Eliminate
C:  20 10^10 (a^x) can give unit digit zero as well as a is equal to x. 
 Correct answer.
We don’t need to look at other options as question is asking about minimum value of ( a+x)

Correct answer C.

vishalkazone · Answer

you misread the question, given a^x is multiple of 20.

CAMANISHPARMAR · Answer

Yes! I agree  vishalkazone  explanation.  garvitbh11  - I hope your doubt is cleared by @ vishalkazone  's explanation. Please feel free to revert in case if you need further clarifications.

generis · Answer

vishalkazone  , please explain what this reference means or why it is inserted:

the exponent  OF  a^x

a  is not the exponent.  a  is the base.

a^{x}  is not the exponent.
 a^{x}  is the term.

"The exponent" is  x

It appears that you omitted the reference or ignored it:

Mathematical workarounds and elision do not answer the question, "If the question intends to target the entire term 'a^x,' then what is the point of mentioning  the exponent  (x)?"

I don't think anyone has answered that question.

vishalkazone · Answer

Many be this can be cleared by question originator, but in my opinion there are two cases:
1- a^x is already exponent so no need to mention exponent. 
2-if it means x is multiple of 20, then it should be written as : exponent of a is multiple of 20, not exponent of a^x is multiple of 20.

stne · Answer

Yes , there seems to be a small typo in the question, the question should state The exponent  a^x is a multiple of 20 rather than: The exponent  of   a^x .
The term  of  is creating all the confusion.

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The exponent of a^x is a multiple of 20, what is the minimum p

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The exponent of a^x is a multiple of 20, what is the minimum p

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