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1.) What is the remainder when 5 to the 999,999th power is divided by 7?

 

2.) A palindrome is a number that reads the same forward and backward. The largest possible 4-digit palindrome in base 3 can be expressed as a 3-digit palindrome in another base b>2. What is b?

 

tysmm

 Jun 8, 2021
 #1
avatar+1893 
+2

2.) A palindrome is a number that reads the same forward and backward. The largest possible 4-digit palindrome in base 3 can be expressed as a 3-digit palindrome in another base b>2. What is b?

The largest 4 digit palindrome in base 3 is 2222. 

Trying all the bases, in base 6, 2222 (base 3) = 212 (base 6). 

 

=^._.^=

 Jun 8, 2021
 #2
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0

ohhh i didnt think of that! TYSMM

Guest Jun 8, 2021
 #4
avatar+1893 
+2

You're welcome. :))

As for the first one, try using mod 7. 

5, 25, 125, 625

5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1...

So it repeats every 6. 

Can you take it from here?

 

=^._.^=

catmg  Jun 8, 2021
 #3
avatar+120075 
+3

For  n ≥  1

 

5 ^(3*2n) mod  7 =   1

 

5^(3(2n + 1) )  mod 7  =  6

 

5^(3 * (2 * 166666 + 1) )  mod 7  =  6

 

5^ ( 3  * 333333)  mod 7  =  6

 

5^(999999)  mod 7   =  6

 

 

cool cool cool

 Jun 8, 2021
 #5
avatar+24 
0

thank you i got it!! :)

akacindy  Jun 8, 2021

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