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

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

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..

Oh , i have misunderstood the question ...
but why not saying x is a multiple of 20 ..

... 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

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

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?

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

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.

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

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.

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

the exponent OF a^x [is a multiple of 20]

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.

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.

Bunuel
could you prove an answer to this question?

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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