Difference between revisions of "2007 iTest Problems/Problem 6"
(New page: ==Problem== Find the units digit of the sum <cmath>\sum_{i=1}^{100}(i!)^{2}</cmath> <math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\...) |
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==Solution== | ==Solution== | ||
If i is less than 5, then <math>i!</math> has a positive units digit, if <math>i\geq 5</math>, then <math>i!</math> has a units digit of 0, as does <math>(i!)^2</math>. So we only need to worry about i=1-4. | If i is less than 5, then <math>i!</math> has a positive units digit, if <math>i\geq 5</math>, then <math>i!</math> has a units digit of 0, as does <math>(i!)^2</math>. So we only need to worry about i=1-4. | ||
+ | *<math>(1!)^2=1</math> | ||
+ | *<math>(2!)^2=4</math> | ||
+ | *<math>(3!)^2=36</math> | ||
+ | *<math>(4!)^2=576</math> | ||
+ | *<math>1+4+6+6=17</math>, which has a units digit of 7 <math>\Rightarrow \mathrm{(E)}</math> | ||
− | + | ==See Also== | |
+ | {{iTest box|year=2007|num-b=5|num-a=7}} | ||
− | + | [[Category:Introductory Number Theory Problems]] | |
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Latest revision as of 19:45, 14 January 2008
Problem
Find the units digit of the sum
Solution
If i is less than 5, then has a positive units digit, if , then has a units digit of 0, as does . So we only need to worry about i=1-4.
- , which has a units digit of 7
See Also
2007 iTest (Problems, Answer Key) | ||
Preceded by: Problem 5 |
Followed by: Problem 7 | |
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