# Homework Help: Math olympiad number theory

1. May 30, 2017

### vishnu 73

• Member warned that some attempt must be shown
1. The problem statement, all variables and given/known data
this problem came out in the math olympiad i took today and i got completely wrecked by this

consider the following equation where m and n are positive integers:

3m + 3n - 8m - 4n! = 680
determine the sum all possible values of m:

2. Relevant equations
not sure which

3. The attempt at a solution
no attempt my mind is completely blank give me some hints please

2. May 30, 2017

### scottdave

Try moving everything with m to one side and n to the other side. What can you say about the values of the m-function and n function ?

3. May 30, 2017

### BvU

Were you allowed to use a calculator ?
Code (Text):

n: 1 2 3 4 5 6
m 1 -6 -4 -2 -20 -242 -2156
2 -8 -6 -4 -22 -244 -2158
3 2 4 6 -12 -234 -2148
4 48 50 52 34 -188 -2102
5 202 204 206 188 -34 -1948
6 680 682 684 666 444 -1470

4. May 30, 2017

### vishnu 73

5. May 30, 2017

### BvU

Just kidding. But from the brute force results you should be able to extract some restrictions on the (m,n) pairs to check. Especially in combination with a list of powers of 3:
3
9
27
81
243
729

and a list of 4n! helps a lot too!
4
8
24
96
480
2880
These don't take long to set up, even without a calculator.

subsequent hypotheses:
• one of the two has to be 6
• m has to be 6, n has to be smaller than 5

I think the hard part is to find the one (6,1) combination that does yield 680. Then 'conclude' that there can be no others.

6. May 30, 2017

### vishnu 73

@BvU
not to offend you but i am not really fan of brute force approaches

and why do you say n has to be 6

based on what scottdave said

i inferred m > n is this true
and if it is does it help to write m = k + n

7. May 30, 2017

### Buffu

$3^m + 3^n = 680 + 8m + 4n!$ or $3^m + 3^n = 4(170 + 2m + n!)$

Since $170 + 2m + n!$ is divisible by $3$ what does this tell you about parity of $170 + 2m + n!$ ? and for what values of $n$ it is possible ?

8. May 30, 2017

### BvU

No offence taken. I'm just interested, and to be sure: I don't know the proper way to deal with this one, so I just did trial and error. 4n! goes real fast as you can see, so n can't be 6. You need at least one of the two with a value of 6 to get over 680 .

I hadn't pulled out the factor 4 Buffu mentions. Hats off !

9. May 30, 2017

### vishnu 73

oh wow that is interesting so 2m + n! ≡ 1 mod3

if n≥3 then m must be multiple of 3

if n <3 then 2m ≡ 2 mod 3

am i getting somewhere

10. May 30, 2017

### vishnu 73

@BvU
and more over i am a really careless boy i am sure to make lots of mistakes i probably lost 8 questions just by careless today so that's another reason i did not opt for your method without a calculator so ya anyways trial and error to estimate bounds and restrictions is always uselful

11. May 30, 2017

### Buffu

You can do what you are doing but an easy way would be to check if $170 + 2m + n!$ is odd or even ?

12. May 30, 2017

### vishnu 73

oh is n't that always even for n bigger than 2

but what does that tell you the left hand side is also always even

13. May 30, 2017

### Buffu

But we want it odd, so what values of $n$ is available to us ?

14. May 30, 2017

### vishnu 73

why do we want it odd? please i am beginner in olympiads

edit :

oh sorry now i get why it must be odd because it is divisible by 3 sorry for being so dumb

so n= 1 to give 170 + 8m + 1! to be odd

3m = 680 + 8m + 1

so m must be at least 6 and so is it only m = 6 solves it

that is very brilliant method there how did you get to it did you just play around with the expression until you got something useful

Last edited: May 30, 2017
15. May 30, 2017

### Buffu

Nooo, It is wrong, since even multiples of 3 are there :).

I said that expression is odd because $3^n + 3^m$ is not divisible by 8 for any value of $m,n$ * and if $170 + 8m + n!$ is even, then $3^m + 3^n$ must be divisible by $8$.

* $3^n \mod 8 = 1, 3$ using power rule for modulus and the fact that $3^n = 1$ for $n = 0$.

So $3^n + 3^m \mod 8 = 2, 4, 6$ hence never divisible by 8.

16. May 30, 2017

### Buffu

I remember some tricks from olympiads when I used to give those.

17. May 31, 2017

### vishnu 73

oh okay now i get it thanks for the help