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Last Digit of a Number

```
Date: 04/14/97 at 19:18:36
From: Hershel Raff
Subject: Last Digit of a Number

What is the last digit in (1996^1997)-(1997^1996)?  I tried to use
examples with smaller numbers and did not get a consistent answer.  I
tried logs:  logY = 1997log1996-1996log1997 and the solution is a
mess.  HELP!
```

```
Date: 04/14/97 at 22:04:38
From: Doctor Steven
Subject: Re: Last Digit of a Number

Look at only the remainders of the numbers when divided by 10.
Notice 6 to any power (except 0) ends in 6. So 1996^1997 ends in a 6.
The question is what does 1997^1996 end in.  Look at what 7^1996 ends
in.  Notice:

7^1 =     7 leaves a remainder of 7 when divided by 10
7^2 =    49 leaves a remainder of 9 when divided by 10
7^3 =   343 leaves a remainder of 3 when divided by 10
7^4 =  2401 leaves a remainder of 1 when divided by 10
7^5 = 16807 leaves a remainder of 7 when divided by 10

Notice we're back to a 7 in the remainder value after the fifth power.
This pattern continues, so every fifth power of 7 ends in 7. In fact
this property is true for every integer (for example, 17^5 ends in a
7, 13^5 ends in a 3).

1997^1996 = 1997^1995 * 1997.  Look at the last digit to get that it
ends with the same digit as 7^1995 * 7.  7^1995 = (7^399)^5 * 7, so
this ends in the same digits as 7^399 * 7 = 7^400 = (7^80)^5 ends in
the same digit as 7^80 = (7^16)^5 so this ends in the same digit as
7^16 = 7^15 * 7 = (7^3)^5*7 ends in the same digit as 7^3 * 7 = 7^4 =
2401, so it ends in 1.

So 1996^1997 ends in 6 and 1997^1996 ends in 1, so
1996^1997 - 1997^1996 ends in 6 - 1 = 5.

Hope this helps.

-Doctor Steven,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 04/15/97 at 20:03:53
From: Hershel Raff
Subject: Re: Last Digit of a Number

Thank you so much for your help.  I was really close on but your
solution worked great!  What a great service!

Hershel Raff
```
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