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Difference between revisions of "1961 AHSME Problems/Problem 28"
(→‎SOLUTION 2)
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==SOLUTION 2==
 
==SOLUTION 2==
  
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* <math>Lemma (Fermat's Theorem)</math>: If <math>a</math> is an integer and <math>p</math> is a prime that is prime to a the we have <math>a^p-1</math> \equiv 1 (<math>mod</math> <math>p</math>).
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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).
  
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Let's define <math>U(</math>x<math>)</math> as units digit funtion of <math>x</math>.
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Let's define </math>U(<math>x</math>)<math> as units digit funtion of </math>x<math>.
 
We can clearly observe that,
 
We can clearly observe that,
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<math>U(</math>7^1<math>)</math>= <math>7</math>
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</math>U(<math>7^1</math>)<math>= </math>7<math>
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     <math>.</math>
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     </math>.
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    <math>.</math>
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    .
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    <math>.</math>
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    .<math>
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<math>U(</math>7^4)<math>= </math>1<math>
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</math>U(<math>7^4)</math>= <math>1</math>
  
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and we can see by Fermat's Theorem that this cycle repeats with the cyclicity of </math>4<math>. Now </math>753<math> = </math>4k + 1<math>  \Rightarrow </math>U(<math>7^753</math>)<math> =</math>7<math>.
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and we can see by Fermat's Theorem that this cycle repeats with the cyclicity of <math>4</math>. Now <math>753</math> = <math>4k + 1</math>  \Rightarrow <math>U(</math>7^753<math>)</math> =<math>7</math>.
 
   
 
   
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   </math> ~GEOMETRY-WIZARD $
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   <math> ~GEOMETRY-WIZARD </math>
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==See Also==
 
==See Also==

Revision as of 08:49, 31 December 2023

Problem 28

If $2137^{753}$ is multiplied out, the units' digit in the final product is:

$\textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 9$

Solution

$7^1$ has a unit digit of $7$. $7^2$ has a unit digit of $9$. $7^3$ has a unit digit of $3$. $7^4$ has a unit digit of $1$. $7^5$ has a unit digit of $7$.

Notice that the unit digit eventually cycles to itself when the exponent is increased by $4$. 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 $753$ leaves a remainder of $1$ after being divided by $4$, the units digit of $2137^{753}$ is $7$, which is answer choice $\boxed{\textbf{(D)}}$.

SOLUTION 2

  • $Lemma$ ($Fermat's$ $Theorem$): If $p$ is a prime and $a$ is an integer prime to $p$ then we have $a^p-1 \equiv 1\ (\textrm{mod}\ p).

Let's define$ (Error compiling LaTeX. Unknown error_msg)U($x$)$as units digit funtion of$x$. We can clearly observe that,$U($7^1$)$=$7$$ (Error compiling LaTeX. Unknown error_msg). 

    .
    .$$ (Error compiling LaTeX. Unknown error_msg)U($7^4)$= $1$

and we can see by Fermat's Theorem that this cycle repeats with the cyclicity of $4$. Now $753$ = $4k + 1$ \Rightarrow $U($7^753$)$ =$7$. 

 $~GEOMETRY-WIZARD$

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

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