Difference between revisions of "1999 AMC 8 Problems/Problem 24"
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==Solution== | ==Solution== | ||
− | Note that the units digits of the powers of 9 have a pattern:<math>9^1 = {\bf 9}</math>,<math>9^2 = 8{\bf 1}</math>,<math>9^3 = 72{\bf 9}</math>,<math>9^4 = 656{\bf 1}</math>, and so on. Since all natural numbers with the same last digit have the same remainder when divided by 5, the entire number doesn't matter, just the last digit. For even powers of <math>9</math>, the number ends in a <math>1</math>. Since the exponent is even, the final digit is <math>1</math>. Note that all natural numbers that end in <math>1</math> have a remainder of <math>1</math> when divided by <math>5</math>. So, our answer is <math>\boxed{1}</math>. | + | Note that the units digits of the powers of 9 have a pattern: <math>9^1 = {\bf 9}</math>,<math>9^2 = 8{\bf 1}</math>,<math>9^3 = 72{\bf 9}</math>,<math>9^4 = 656{\bf 1}</math>, and so on. Since all natural numbers with the same last digit have the same remainder when divided by 5, the entire number doesn't matter, just the last digit. For even powers of <math>9</math>, the number ends in a <math>1</math>. Since the exponent is even, the final digit is <math>1</math>. Note that all natural numbers that end in <math>1</math> have a remainder of <math>1</math> when divided by <math>5</math>. So, our answer is <math>\boxed{1}</math>. |
Revision as of 13:21, 17 June 2011
Problem 24
When is divided by
, the remainder is
Solution
Note that the units digits of the powers of 9 have a pattern: ,
,
,
, and so on. Since all natural numbers with the same last digit have the same remainder when divided by 5, the entire number doesn't matter, just the last digit. For even powers of
, the number ends in a
. Since the exponent is even, the final digit is
. Note that all natural numbers that end in
have a remainder of
when divided by
. So, our answer is
.