+0  
 
+1
150
6
avatar+484 

How many integers $n$ satisfy $0

 Apr 6, 2021
edited by RiemannIntegralzzz  Apr 6, 2021
 #1
avatar+484 
+1

Sorry, it is not rendering for some reason.

 

Here is the question without Latex:

 

How many integers n satisfy 0 is less than n is less than 60 and 4n=2 (mod 6)

 Apr 6, 2021
edited by RiemannIntegralzzz  Apr 6, 2021
edited by RiemannIntegralzzz  Apr 6, 2021
edited by RiemannIntegralzzz  Apr 6, 2021
edited by RiemannIntegralzzz  Apr 6, 2021
edited by RiemannIntegralzzz  Apr 6, 2021
 #2
avatar+149 
+2

0 < n < 60

\(4n\equiv 2\pmod 6\)

\(n\equiv \frac{2}{4}\pmod6\)

\(n\equiv \frac{2+6}{4}\equiv2\pmod6\)

n = 6k + 2

k = 0, 1, 2, ..., 9

total : 10

Jamesaiden  Apr 6, 2021
edited by Jamesaiden  Apr 6, 2021
 #3
avatar+484 
+1

Sorry, it was incorrect. I have one more try however.

RiemannIntegralzzz  Apr 6, 2021
 #4
avatar+114070 
+3

My attempt

All pronumerals are integers

 

LaTex not displaying so here is a pic that I took of it.

 

 

-----------------------------------------------------------------------------------------------------------------

 

\(0

 

which means there are 18 possible vaues of g and hence 18 possible values of n

 

 

LaTex:

0 4n=2\mod6\\
2n=1\mod3\\
2n=3k+1\\
2n-3k=1\\
\text{One solution is n=2, k=1 }\\
2(2\qquad)-3(1\qquad)=1\\
2(2+3g)-3(1+2g)=1\\
so\\
n=2+3g\\
0<2+3g<60\\
-2<3g<58\\
-0.6 0\le g\le17

 Apr 6, 2021
edited by Melody  Apr 6, 2021
 #5
avatar+484 
+3

Sorry @Melody and @Jamesaiden, they were both wrong. The correct answer was 20. Here was the solution they gave:

 

The residue of 4n(mod 6) is determined by the residue of n (mod 6). We can build a table showing the possibilities(it doesn't render, but here it is with words)

 

n (mod 6)             0       1        2           3          4          5                                                          

----------------------------------------------------------------------------------------------------

4n (mod 6)           0        4       2           0          4          2                                               

 

 

As the table shows, 4n=2 (mod 6) is true when n=2 or n=5 (mod 6). Otherwise, it's false.

 

So, our problem is to count all n between 0 and 60 that leave a remainder of 2 or 5 (mod 6). These integers are 2, 5, 8, 11, 14, 17, ... , 56, 59.

 

There are $\boxed{20}$ integers in this list.

 

 

Thank you both for trying, however! :)

RiemannIntegralzzz  Apr 6, 2021
edited by RiemannIntegralzzz  Apr 6, 2021
 #6
avatar+114070 
+1

I just made a careless mistake.

 

58/3 = 19.3333...    NOT   17.33333

 

So I also got 20 values of n

 Apr 6, 2021

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