Last Digit of a NumberDate: 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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