Problem Solving

If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18

OA[spoiler]:C[/spoiler]

Help me solving the problem

kumar720 wrote:If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18

OA[spoiler]:C[/spoiler]

Help me solving the problem

i go with C as well...

30/3 + 30/9 + 30/27

10 + 3 + 1 = 14

hence C .

hope that helps...

Did the original question asked k when 3^k?

or was it 3*k?

Hi Manulath, it was 3^k, typo..

sudhir3127 wrote:

kumar720 wrote:If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18

OA[spoiler]:C[/spoiler]

Help me solving the problem

i go with C as well...

30/3 + 30/9 + 30/27

10 + 3 + 1 = 14

hence C .

hope that helps...

Can you please specify your logic?
You mean that 30! has 10 multiplies of 3, 2 multiplies of 9 and 1 of 27?

4meonly wrote: Can you please specify your logic?
You mean that 30! has 10 multiplies of 3, 2 multiplies of 9 and 1 of 27?

That is exactly what Sudhir did.

Essentially, what you want to do here is determine how many times 3 is a factor of 30!. The most foolproof way of doing this just to write it out:

30: 3x10
27: 3x3x3
24: 3x8
21: 3x7
18: 3x3x2
15: 3x5
12: 3x4
9: 3x3
6: 3x2
3: 3x1

Count up your threes, and you'll see 14 of them, so the answer is C.

Hope this helps!

Can someone explain what and how Sudir did what he did

sumithshah wrote:Can someone explain what and how Sudir did what he did

That is a method to calculate what highest power of a number will divide a factorial.

For eg, what highest power of 2 will divide 30!

So, what we do is take all powers of 2, which are small than the factorial number ( in this case 30 )

So, we have powers of 2 as (2 , 4 , 8, 16 ) Note - its till 16 because 32 is greater than 30

Now, we divide 30 by each power and take the sum of quotient of the division

so 30/ 2 = 15
30/4 = 7
30/8 = 3
30/16 = 1

where (15, 7, 3, 1) are quotient of the division.

Now, take the sum of these, which is 26. Hence, highest power of 2 that divides 30! is 24

Hope it was helpful

u bet it was! someone should make a document with all these shourtcuts. Until now I was writing down all events from 1-30 and all that jazz!

nitin86 wrote:

sumithshah wrote:Can someone explain what and how Sudir did what he did

That is a method to calculate what highest power of a number will divide a factorial.

For eg, what highest power of 2 will divide 30!

So, what we do is take all powers of 2, which are small than the factorial number ( in this case 30 )

So, we have powers of 2 as (2 , 4 , 8, 16 ) Note - its till 16 because 32 is greater than 30

Now, we divide 30 by each power and take the sum of quotient of the division

so 30/ 2 = 15
30/4 = 7
30/8 = 3
30/16 = 1

where (15, 7, 3, 1) are quotient of the division.

Now, take the sum of these, which is 26. Hence, highest power of 2 that divides 30! is 24

Hope it was helpful

If I apply this method to the above question for the powers of 3 I get (10, 3, 9). Sum of these is 22. How would I arrive at 14?

I posted an explanation here:

https://www.beatthegmat.com/product-of-t ... 94247.html

kumar720 wrote:If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18

OA[spoiler]:C[/spoiler]

Help me solving the problem

p is the product of the integers from 1 to 30 means P is the factorial of 30....

p = 30!


Greatest integer k for which 3k is a factor of p means - the number must contain 3 and a product of k.

So 30! / 3 = 10

10/3 = 3

3/3 = 1

Total 10 + 3 + 1 = 14...

nitin86 wrote:

sumithshah wrote:Can someone explain what and how Sudir did what he did

That is a method to calculate what highest power of a number will divide a factorial.

For eg, what highest power of 2 will divide 30!

So, what we do is take all powers of 2, which are small than the factorial number ( in this case 30 )

So, we have powers of 2 as (2 , 4 , 8, 16 ) Note - its till 16 because 32 is greater than 30

Now, we divide 30 by each power and take the sum of quotient of the division

so 30/ 2 = 15
30/4 = 7
30/8 = 3
30/16 = 1

where (15, 7, 3, 1) are quotient of the division.

Now, take the sum of these, which is 26. Hence, highest power of 2 that divides 30! is 24

Hope it was helpful

Why would the highest power of 2 be 24 and not 26?

Thanks for your help

Poisson wrote:

nitin86 wrote:

sumithshah wrote:Can someone explain what and how Sudir did what he did

That is a method to calculate what highest power of a number will divide a factorial.

For eg, what highest power of 2 will divide 30!

So, what we do is take all powers of 2, which are small than the factorial number ( in this case 30 )

So, we have powers of 2 as (2 , 4 , 8, 16 ) Note - its till 16 because 32 is greater than 30

Now, we divide 30 by each power and take the sum of quotient of the division

so 30/ 2 = 15
30/4 = 7
30/8 = 3
30/16 = 1

where (15, 7, 3, 1) are quotient of the division.

Now, take the sum of these, which is 26. Hence, highest power of 2 that divides 30! is 24

Hope it was helpful

Why would the highest power of 2 be 24 and not 26?

Thanks for your help

Bumping for clarification. Thanks again