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I believe the answer is 27.

For “n” to be an integer, each factorial expression must be divisible by 3^x.

Specifically, 59! has to be divisible by 3^x.

Therefore, the problem now boils down to figuring out how many factors of 3 exist in 59!.

By my count, the number is 27.

Good luck!

My answer is 27.

Numerator = 59! + 60! + 61! = 59! * (1 + 60 + 60 * 61) = 59! * (61 + 60 * 61) = 59! * (61 * (1 + 60)) = 59! * 61^2

Since 61^2 cannot be divided by 3, so the problem reduces to n = 59! / 3^x, that is how many 3s are there in 59!.

59/3 = 19 … 2, 19/3 = 6 … 1, 6/3 = 2. So the answer is 19 + 6 + 2 = 27.

is it 19?

59

Is the answer 27?

Sorry, it’s 27

Ans : 27

As we can write (61!+60!+59!) = 59!(61×60 + 60 + 1) = 61x61x59!

And in 59!, the highest power of 3 is 27.

Hmm…I think you got the right answer, but I’m not sure. The question isn’t asking for thie highest power of ‘3’, but how many 3’s can you factor from 61! + 60! + 59!. That also happens to be 27. But maybe that’s what you meant 🙂

Answer is 26

For finding the powers of a prime number p, in the n! The formula is: (n/p)+(n/p^2)+….. (n/p^k)

, where k must be chosen such that p^k <= n

Solving for 59!

(59/3)+(59/9)+(59/27)=19+6+2=27

Therfore 59! Will have 27 powers of 3 or 3^27

Similarly solving for 60! And 61! We get 3^29 and 3^29 respectively.

Now 'n' will have (3^26+3^29+3^29)/3^x

So maximum value of x for n to be an integer is 26

Answer is 27

For finding the powers of a prime number p, in the n! The formula is: (n/p)+(n/p^2)+….. (n/p^k)

, where k must be chosen such that p^k <= n

Solving for 59!

(59/3)+(59/9)+(59/27)=19+6+2=27

Therfore 59! Will have 27 powers of 3 or 3^27

Similarly solving for 60! And 61! We get 3^29 and 3^29 respectively.

Now 'n' will have (3^27+3^29+3^29)/3^x

So maximum value of x for n to be an integer is 27

Yep, that’s it! Perfect 🙂

n = 59!*( 1 + 60 + (60*61) ) / 3^x

n= 59! *(3721) / 3^x

Now, as 3721 is not divisible by 3 , it won’t be divisible by any of its higher powers.

Hence, the answer should be equal to the highest power of 3 in 59!, which is 27.

Hi Siddharth,

You got the right answer, but an even quicker way is to see that (1 + 60 + 60*61) = 61^2, and 61 is not divisible by ‘3’.

Good job!

Is the answer 22?

Almost, but make sure to include the 3’s in 27 and 54.

22

If we simplify this it looks something like

(61*60*59! + 60*59! + 59!) factor out 59!:

59!(61*60*1 +60 + 1) = 59!(61*60 + 61) factor out 61:

59!*61*(60+1) so that’s 59!*61*61

the 61^2 will not play a roll in divisibility by 3^x because 61 is not divisible 3.

so we have to focus on 59!

lets expand to see if we can find a pattern.

59! = 1*2*(3)*4*5*(3*2)*7*8*(3*3)*10*11*(3*4)

notice that every third number has atleast one 3. so how many 3’s are in 59!?

59/3 =19 so thats 19 positions with atleast one 3. and then we have to look for the actual exponents of 3 so for 9 theres 1 extra 3 in that position and for 27 there are 2 extra 3s. so total of 22 threes will factor in to 59!61^2

Hi Urvish,

Almost, but don’t forget the multiples of ‘9’ and ’27’ that are less than 59 🙂

22

If we simplify this it looks something like

(61*60*59! + 60*59! + 59!) factor out 59!:

59!(61*60*1 +60 + 1) = 59!(61*60 + 61) factor out 61:

59!*61*(60+1) so that’s 59!*61*61

the 61^2 will not play a roll in divisibility by 3^x because 61 is not divisible 3.

so we have to focus on 59!

lets expand to see if we can find a pattern.

59! = 1*2*(3)*4*5*(3*2)*7*8*(3*3)*10*11*(3*4)

notice that every third number has atleast one 3. so how many 3’s are in 59!?

59/3 =19 so thats 19 positions with atleast one 3. and then we have to look for the actual exponents of 3 so for 9 theres 1 extra 3 in that position and for 27 there are 2 extra 3s. so total of 22 threes will factor in to 59!61^2

umm.. answer is 26?

edit-

explaination:-

simplifying. the numerator

59!(61X60+60+1)=59!X61X61=59X58X57!X61X61

now only term 57! is divisible by 3

now the no.of 3’s needed can be calculated as

greatest no. in 57! which is divisible by 3 is 57 itself

3. ,19’s are 57 therefore, there are total 19 no.s that are divisible by 3 of which 6 no.s are divisible by 9(as 9X6=54 is largest in it ) , and in these no’s one no. is 27 .Now total no.of 3’s from 3,9 and 27 are (19-6)X1+(5X2)+(1X3) = 26. as 3^1=3 ,3^2=9 and 3^3=27 Hence 3^26 is the larget required no. [may be]

p.s- I know procedure is little long , im looking forward to your short explaination

Hi Tarun,

Sooo close. I think you missed one of the 3’s. The answer is 27. I think you forgot the three 3’s in 54. I’ll have a quick answer up later this week 🙂

haha yeah dang!! I missed 27 in 54 that would make it 27

The answer is 27….)

59! (60*61 + 60 + 1) = 59!*61*61. number of 3’s in this is = number of multiples of 3 (19) + extra 3’s in multiples of 9 and 27(6+2=8) .Total 27 3’s. x= 27

Hi Chris,

I think x=27.

Yep, that’s it!

27 should be the answer. First set of multiples of 3 in 59!*61*61 are 3 through 57, 19 in total, then from 3 to 18, 6 in total and then 3 and 6, 2 in total. Adding them results in 27

I think it’s 22. The factorial can be simplified as 59!*61*61. Now since 59! Has 19 multiples of 3 in it and 3 more for 27 and 9. Thus 22 in total.

Is it 19? I think the factorial can be simplified into the form 59!*61^2. Now since 59! Has 20 numbers in it that are multiples of 3, the answer should be 19.

I presume that the standard formula to determine prime decomposition should help. Using the same, amongst (61!+60!+59!) for n to be an integer, its suffices to consider the smallest amongst the 3 values, that is, 59!/3^x which works out to be 59/3 + 59/3^2 + 59/3^3 = 19+6+2 = 27 (taking integer values)

Yep, that’s it! Good job 🙂

hi Chris,

value of x = 27.

59/3 + 59/9 + 59/27 =27.

Good–you got it! Of course factoring out of 61 is important as well 🙂

Hello Chris,

you are doing a great job with your posts and helping thousands of students

i think the answer is 26

(61!+60!+59!)/3^x,

The numerator is 59! (61*61*61)

59! will consist of 19 multiples of 3 that are starting from 57 till 3

So, x will be 1+3+……19= 19*20/2=190

Greatest value can be 190

umm.. answer is 26?

explaination:-

simplifying. the numerator

59!(61X60+60+1)=59!X61X61=59X58X57!X60X61

now only term 57! is divisible by 3

now the no.of 3’s needed can be calculated as

greatest no. in 57! which is divisible by 3 is 57 itself

3. ,19’s are 57 therefore, there are total 19 no.s that are divisible by 3 of which 6 no.s are divisible by 9(as 9X6=54 is largest in it ) , and in these no’s one no. is 27 .Now total no.of 3’s from 3,9 and 27 are (19-6)X1+(5X2)+(1X3) = 26. as 3^1=3 ,3^2=9 and 3^3=27 Hence 3^26 is the larget required no. [may be]

p.s- I know procedure is little long , im looking forward to your short explaination

x should be 19

My logic was that 59! needs to be dived by a 3^x such that there no remainders. 57 is the largest number in 59! that is divisible by 3 which is 3*19.

This is applicable for 60! and 61! Cant have 20 3’s as that may satisfy 60! and 61! but not 59! so n would not remain and integer.

Is it 27 ?

The greatest value of x is 27.

For n to be integer, 61!/3^x, 60!/3^x and 59!/3^x all the three terms should be integers.

And we can find the greatest value of x for which 59!/3^x be an integer then obviously the other two terms will also be integers..

And there are 27 threes in 59!.

So the ans should be 27

For n to be integer, 61!/3^x , 60!/3^x & 59!/3^x …all the three terms should be an integer.

And if we can the greatest value of x for 59!/3^x to be integer , obviously the other two terms will also be integer…

And there are 27 threes in 59!

So the answer is 27..

Is it 27 ??

If it is right then I will tell the explanation.

27 is answer.

The expression can be simplified as 61^2*59!/3^x .

The max value of x will be number of 3s in 59! which is 27. Bravo.

I think x will be 26….