Difference between revisions of "1961 AHSME Problems/Problem 28"
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* <math>Lemma</math> (<math>Fermat's</math> <math>Theorem</math>): If <math>p</math> is a prime and <math>a</math> is an integer prime to <math>p</math> then we have <math>a^{p-1} \equiv 1\ (\textrm{mod}\ p)</math>. | * <math>Lemma</math> (<math>Fermat's</math> <math>Theorem</math>): If <math>p</math> is a prime and <math>a</math> is an integer prime to <math>p</math> then we have <math>a^{p-1} \equiv 1\ (\textrm{mod}\ p)</math>. | ||
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*Let's define <math>U</math>(<math>x</math>) as units digit funtion of <math>x</math>. | *Let's define <math>U</math>(<math>x</math>) as units digit funtion of <math>x</math>. | ||
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<math>U</math>(<math>7^4)= </math>1<math> | <math>U</math>(<math>7^4)= </math>1<math> | ||
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− | </math>U(<math>7</math><math>^753</math>)<math> = </math>7<math>. | + | |
+ | </math>and<math> </math>we<math> </math>can<math> </math>see<math> </math>by<math> </math>Fermat's<math> </math>Theorem<math> </math>that<math> </math>this<math> </math>cycle<math> </math>repeats<math> </math>with<math> </math>the<math> </math>cyclicity<math> of </math>4<math>. </math>Now<math> </math>753<math> = </math>4k<math> + </math>1<math> \Rightarrow </math>U(<math>7</math><math>^753</math>)<math> = </math>7<math>. | ||
</math> ~GEOMETRY-WIZARD $ | </math> ~GEOMETRY-WIZARD $ |
Revision as of 08:59, 31 December 2023
Contents
Problem 28
If is multiplied out, the units' digit in the final product is:
Solution
has a unit digit of
.
has a unit digit of
.
has a unit digit of
.
has a unit digit of
.
has a unit digit of
.
Notice that the unit digit eventually cycles to itself when the exponent is increased by . It also does not matter what the other digits are in the base because the units digit is found by multiplying by only the units digit. Since
leaves a remainder of
after being divided by
, the units digit of
is
, which is answer choice
.
SOLUTION 2
(
): If
is a prime and
is an integer prime to
then we have
.
- Let's define
(
) as units digit funtion of
.
We can clearly observe that,
(
)=
. . . . . .
(
1$$ (Error compiling LaTeX. Unknown error_msg)and$$ (Error compiling LaTeX. Unknown error_msg)we$$ (Error compiling LaTeX. Unknown error_msg)can$$ (Error compiling LaTeX. Unknown error_msg)see$$ (Error compiling LaTeX. Unknown error_msg)by$$ (Error compiling LaTeX. Unknown error_msg)Fermat's$$ (Error compiling LaTeX. Unknown error_msg)Theorem$$ (Error compiling LaTeX. Unknown error_msg)that$$ (Error compiling LaTeX. Unknown error_msg)this$$ (Error compiling LaTeX. Unknown error_msg)cycle$$ (Error compiling LaTeX. Unknown error_msg)repeats$$ (Error compiling LaTeX. Unknown error_msg)with$$ (Error compiling LaTeX. Unknown error_msg)the$$ (Error compiling LaTeX. Unknown error_msg)cyclicity
4
Now$$ (Error compiling LaTeX. Unknown error_msg)753
4k
1
U(
)
7
~GEOMETRY-WIZARD $
See Also
1961 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.