+0

# What are the last two digits of 5^2015

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362
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What are the last two digits of 5^2015

Guest May 19, 2015

#2
+14536
+19

### Here are all digits !

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#1
+5

25 detail detail detail detail

Guest May 19, 2015
#2
+14536
+19

### Here are all digits !

#3
+675
+3

∞infinity∞infinity∞infinity∞infinity∞infinity∞infinity∞infinity∞infinity∞infinity∞infinity∞infinity∞infinity∞infinity∞infinity∞infinity∞infinity

thejamesmachine  May 19, 2015
#4
+92164
+5

5^2015

\$\$\\5^1=5\\
5^2=25\\
5^3=125\\
5^4=525\\\$\$

It seems that if the power is bigger than 1 that the last 2 digiits will always be 25

MAYBE SOMEONE WOULD LIKE TO PROVE THIS  ??

Melody  May 20, 2015
#5
+85585
+10

Here's a "proof"

5^2 = 25

5^3  =  25 * 5  =  (20 + 5) (5) = 100 + 25

5^4  = 125 * 5  = (120 + 5) (5)  = (100 + 20 + 5) (5) = 500 + 100 + 25

5^5 = 625 * 5 =  (600 + 20 + 5) (5)  = (500 + 100 + 25)(5) = (500 + 100 + 20 + 5) (5)  = 2500 + 500 + 100 + 25

So it appears that the pattern  for 5^n =

25 + 100*5^(0) + 100*5^(1) +.....+ 100*5^(n - 4) + 100*5^(n- 3)  for n ≥ 5

And, in this series, all terms except the first one will end in the digits "00"...so adding 25 to this sum will always result in a number ending in "25"..........

CPhill  May 20, 2015
#6
+92164
0

Thanks CPhill

Melody  May 21, 2015

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