M15-15

Question

If  x  is the sum of factorials of the first  n  positive integers, where  n > 0  (i.e. for  n =3 ,  x=1!+2!+3! ), what is the units digit of  x ?

(1)  n  is divisible by 4
 
(2)  \frac{n^2 + 1}{5}  is an odd integer

Bunuel · Accepted Answer

Official Solution:

If  x  is the sum of factorials of the first  n  positive integers (i.e. for  n=3 ,  x=1!+2!+3! ), what is the units digit of  x ? 
 
Generally, the units digit of  1! + 2! + ... +n!  can take ONLY 3 values:
 
A. If  n=1 , then the units digit is 1;
 
B. If  n=3 , then the units digit is 9;
 
C. If  n= \{ 	ext{ANY other value} \} , then the units digit is 3 (if  n=2 , then  1!+2!=3 ; if  n=4 , then  1!+2!+3!+4!=33 , and if  n \ge 4 , then the terms after  n=4  will end by 0 thus will not affect the units digit and it'll remain 3).
 
So basically the question asks whether we can determine which of the three cases we have.
 
(1)  n  is divisible by 4. This statement implies that  n  is not 1 or 3, thus third case. Sufficient.
 
(2)  \frac{n^2 + 1}{5}  is an odd integer. The same here:  n  is not 1 or 3, thus third case. Sufficient.

Answer: D

awal_786 · Answer

For Statement 2, you can also take N=2 which will give (4+1)/5=1 odd integer .. now unit digit will be 3 because of N=2
if take take N=12 again answer will be odd and this time unit digit will be 0
This makes statement 2 insufficient right ?

Bunuel · Answer

Not right.

If N is NOT 1 or 3, then the units digit of  1! + 2! + ... +N!  is 3. 
If N = 2, the units digit is 3 -->  1! + 2!=3 .
If N = 12, the unit digits is 3 too -->  1! + 2! + ... +12!=522,956,313

Bhavanahiremath · Answer

But isnt 1!+2!+...N! where N=1 equal to 4?
1+2+1 = 4
Also when N=2, it becomes 1!+2!+2! = 1+2+2=5, why are we saying that the units digit is 3?
Please let me know where I'm going wrong

Bunuel · Answer

If N = 1, we have only 1! = 1
If N = 2, then we have 1! + 2! = 3
If N = 3, then we have 1! + 2! + 3! = 9
...

Gerry20 · Answer

I think another way to solve this problem would be using the rule below:

Any multiplication series which contains 2*5 will always have units digit as zero.

So just calculate values till 4! after which the unit digit should be the same, as from n=5, all terms from 5! will have units as zero.
 Please see below for clarity:

(n=2) => 1+ 2!=3
(n=3)=> 3+ 3! = 9
(n=4)=> 9+ 4! = 33
After this all the terms that you add will have units digit as zero as they contain 2*5 so units digit is always 3.
(n=5) =>33+5! (contains 2*5) =123 
(n=5) =>33+ 6! (contains 2*5)= 753 and so on.
Hence, both statements are sufficient.

Thanks Bunuel, brilliant question!

Talayva · Answer

Really good question . Loved it

kaladin123 · Answer

I think that this is a  high-quality  question and I agree with the explanation.

SDW2 · Answer

Hello,
Why can't n be zero? if n=0 then there are 0 positive integers and x should be 0!=1 and hence statement 1 should not be sufficient
Please provide clarity on this.  Bunuel   VeritasKarishma   IanStewart

IanStewart · Answer

First, if n were zero, we'd be adding the factorials "of the first zero positive integers". So we'd be adding the factorials of no numbers at all, and 0! doesn't come into it. If you take the simpler sentence "what is the sum of the first zero positive integers", I'm not even sure what a reasonable answer to that question would be -- you could justify the answer "zero", but you could also say the sum doesn't even exist, because we aren't adding anything. The same would be true if you let n = 0 here -- the sum is either zero or doesn't exist; it certainly is not equal to 0! = 1.

In general, in every official question I've seen that says "there are n apples" or "k is the sum of the first n positive integers", it's safe to assume n is positive. That's especially true in this question, because the question arguably doesn't even make sense if n is zero.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Queen776 · Answer

What is this question asking? How many digits are in some of the factorial n? What is units digit, that doesn't even sound like a proper phrase?

Posted from my mobile device

Bunuel · Answer

The units digit of an integer is its last, or right-most, digit. For example, the units digit of 125 is 5. Thus, the question asks about the last digit of the sum of factorials.

M15-15

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