If k is a positive integer and m is the product of the first 40 positi

Question

If k is a positive integer and m is the product of the first 40 positive integers, what is the value of k ?

(1) 10^k is a factor of m.
(2) 10^k is a factor of n, where n is the product of the first 9 positive integers.

niks18 · Accepted Answer

Basically  m=40!

Statement 1:   10^k  is a factor of  40! . but  40!  has  \frac{40}{5}+\frac{40}{25}=8+1=9  trailing  0 s. so  k  could be  1  or  2  or any other number.  insufficient

Statement 2:   n=9!=1*2*3*4*5*6*7*8*9 . this number has only one trailing  0  which will be a multiple of  2*5=10 . hence  k=1 .  Sufficient .

Option  B

rever08 · Answer

From statement 1: k could be 0,1,2.. Insufficient

From statement 2: k could be 0 or 1. Question stem says k is positive. Hence sufficient

Posted from my mobile device

PKN · Answer

Given m=40!
Question stem, k=?

We need only 1 value of k to prove sufficiency & a minimum of 2 values of k to prove insufficiency.

St1:-  10^k is a factor of 40!
40!=1* 2 *3*4* 5 *6*7*8*9* 10 *...*40
So  10^k  is a factor of 10 & 10^2 where k is 1 and 2 respectively.

Insufficient.
st2:- 10^k  is a factor of 9!
9!=1* 2 *3*4* 5 *6*7*8*9
So  10^k  is a factor of 10 when k is 1 only.
Hence, sufficient.

Ans. (B)

durgamadhab · Answer

Ans is B...(1) says k<= 8
(2) k=1

ETLim · Answer

Can we look at it as

1) since there is 10 and 100 in 40!, we unable to determine k
2) there is only 2*5=10 in 9!, hence we can determine k=1

PKN · Answer

Hi  ETLim ,
In your quotation (1), we are able to determine more than one value of 'k'. Hence given statement-1 is insufficient to answer the question stem. Please note: In DS questions those ask the value of variable or things, if we get one value from the statement then that statement is sufficient and if we get more than one value from the statement, then that statement is insufficient. For both st1 & st2 ,individually to be sufficient, we must obtain an unique value from both the statements.

In your quotation (2), since we obtained one value of k, hence st2 is sufficient.

Hope it helps.

chetan2u · Answer

If k is a positive integer and m is the product of the first 40 positive integers, what is the value of k ?

(1) 10^k is a factor of m.
m=1*2*3*.....*40=40!
Number of 10s will depend on the bigger prime factor that is 5.
# of 5s = \frac{40}{5}+\frac{40}{25}= 8+1=9 
So k can take any values till 9 as 10^1, 10^2,....10^9 all are factors of 40!
Insuff

(2) 10^k is a factor of n, where n is the product of the first 9 positive integers.
Now n=9! And number of 5s are 9/5..=1
So k can take Only one value 1
Sufficient

B

adkikani · Answer

chetan2u   niks18   pushpitkc   pikolo2510

How about below analysis of option A?

k\geq{0} 
m = 40!

So m also contains 2*5 ie 10 divisible by 10^1
and m also contains 25*4 which is divisible by 10^2
So k can be 1 or 2.

I could not understand above approach used for examinating statement 1

pushpitkc · Answer

Hi  adkikani

The analysis of statement 1 as already explained by chetan2u is as follows

There is a formula for finding the highest power of a prime number in a factorial
If p is a prime number, then the highest power of p in a factorial n is { \frac{n}{p} } + { \frac{n}{p^2} } + { \frac{n}{p^3} }.....
where { \frac{a}{b} } is the quotient when integer b divides another integer a

Now coming to the problem at hand. Since p is 10, which can be prime-factorized as 2*5, it is
enough if we are able to find the highest power of 5(since it is the biggest prime number). n = 40

Therefore, the highest power of p(10) is nothing but the highest power of 5.
Substituting values in the formula  {\frac{40}{5^1}} + {\frac{40}{5^2}} = {\frac{40}{5}} + {\frac{40}{25}} = 8 + 1 = 9 
This means that the 10^9 will divide 40!. So even the smaller powers of 10 will divide 40!

Hope this clears your confusion!

AkshdeepS · Answer

If the question would have asked for highest value of k, then A would be the answer. Right?

PKN · Answer

I don't think so. Each statement has its own greatest value of k.  (st1(k_{max}=9, st2(k_{max}=1)

In DS value questions, each statement must produce same value. Even answer can't be D.

So, test maker would set question accordingly.

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