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Difference between revisions of "1997 AJHSME Problems/Problem 25"

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==Solution==
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==Solution 1==
  
 
All the tens digits of the product will be irrelevant to finding the units digit.  Thus, we are searching for the units digit of <math>(2\cdot 4\cdot 6 \cdot 8) \cdot (2 \cdot 4 \cdot 6 \cdot 8) \cdot (2\cdot 4\cdot 6 \cdot 8) \cdot ...</math>
 
All the tens digits of the product will be irrelevant to finding the units digit.  Thus, we are searching for the units digit of <math>(2\cdot 4\cdot 6 \cdot 8) \cdot (2 \cdot 4 \cdot 6 \cdot 8) \cdot (2\cdot 4\cdot 6 \cdot 8) \cdot ...</math>
  
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There will be <math>10</math> groups of <math>4</math> numbers.  The number now can be rewritten as <math>(2\cdot 4 \cdot 6 \cdot 8)^{10}</math>
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There will be <math>10</math> groups of <math>4</math> numbers.  The number now can be rewritten as <math>(2\cdot 4 \cdot 6 \cdot 8)^{10}</math>.
  
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Simplifiying the ins384)^{10}<math>
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Multiplying, we get <math>384^{10}</math>.
  
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Again, we can disregard the tens and hundreds digit of </math>384<math>, since we only want the units digit of the number, leaving </math>4^{10}<math>.
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Again, we can disregard the tens and hundreds digit of <math>384</math>, since we only want the units digit of the number, leaving <math>4^{10}</math>.
  
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Now, we try to find a pattern to the units digit of </math>4^n<math>.  To compute this quickly, we once again discard all tens digits and higher.
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Now, we try to find a pattern to the units digit of <math>4^n</math>.  To compute this quickly, we once again discard all tens digits and higher.
  
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</math>4^1 = 4<math>.
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<math>4^1 = 4</math>.
  
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</math>4^2 = 4\cdot 4 = 1\underline{6}<math>, discard the </math>1<math>.
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<math>4^2 = 4\cdot 4 = 1\underline{6}</math>, discard the <math>1</math>.
  
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</math>4^3 = 1 \cdot 4 = \underline{4}<math>
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<math>4^3 = 1 \cdot 4 = \underline{4}</math>
  
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</math>4^4 = 4 \cdot 4 = 1\underline{6}<math>, discard the </math>1<math>.
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<math>4^4 = 4 \cdot 4 = 1\underline{6}</math>, discard the <math>1</math>.
  
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</math>4^5 = 1 \cdot 4 = \underline{4}<math>
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<math>4^5 = 1 \cdot 4 = \underline{4}</math>
  
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Those equalities are, in reality, congruences </math>\mod {10}<math>.
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Those equalities are, in reality, congruences <math>\mod {10}</math>.
  
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Thus, the pattern of the units digits is </math>\{4, 6, 4, 6, 4, 6, 4, 6, 4, 6\}<math>.  The cycle repeats so that term </math>n<math> is the same as term </math>n+2<math>.  The tenth number in the cycle is </math>6<math>, giving an answer of </math>\boxed{D}$
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Thus, the pattern of the units digits is <math>\{4, 6, 4, 6, 4, 6, 4, 6, 4, 6\}</math>.  The cycle repeats so that term <math>n</math> is the same as term <math>n+2</math>.  The tenth number in the cycle is <math>6</math>, giving an answer of <math>\boxed{D}</math>.
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==Solution 2==
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Again, the value we seek is equal to <math>(2\cdot 4\cdot 6\cdot 8)^{10}\mod 10</math>.
 +
We can use equivalence to simplify.
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 +
<math>(2\cdot 4\cdot 6\cdot 8)^{10}\mod 10</math>
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 +
<math>\equiv (2\cdot 4\cdot (-4)\cdot (-2))^{10} \mod 10</math>
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 +
<math>\equiv 64^{10} \mod 10</math>
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<math>\equiv 4^{10} \mod 10</math>
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<math>\equiv 2^{20} \mod 10</math>
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<math>\equiv 8^6\cdot 2^2 \mod 10</math>
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<math>\equiv (-2)^6\cdot 2^2 \mod 10</math>
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<math>\equiv 2^8 \mod 10</math>
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<math>\equiv 256 \mod 10</math>
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<math>\equiv 6 \mod 10</math>
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Thus our answer is <math>\boxed{D}</math>.
  
 
==See Also==
 
==See Also==

Latest revision as of 10:15, 27 June 2023

Problem

All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product?

$\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$


Solution 1

All the tens digits of the product will be irrelevant to finding the units digit. Thus, we are searching for the units digit of $(2\cdot 4\cdot 6 \cdot 8) \cdot (2 \cdot 4 \cdot 6 \cdot 8) \cdot (2\cdot 4\cdot 6 \cdot 8) \cdot ...$

There will be $10$ groups of $4$ numbers. The number now can be rewritten as $(2\cdot 4 \cdot 6 \cdot 8)^{10}$.

Multiplying, we get $384^{10}$.

Again, we can disregard the tens and hundreds digit of $384$, since we only want the units digit of the number, leaving $4^{10}$.

Now, we try to find a pattern to the units digit of $4^n$. To compute this quickly, we once again discard all tens digits and higher.

$4^1 = 4$.

$4^2 = 4\cdot 4 = 1\underline{6}$, discard the $1$.

$4^3 = 1 \cdot 4 = \underline{4}$

$4^4 = 4 \cdot 4 = 1\underline{6}$, discard the $1$.

$4^5 = 1 \cdot 4 = \underline{4}$

Those equalities are, in reality, congruences $\mod {10}$.

Thus, the pattern of the units digits is $\{4, 6, 4, 6, 4, 6, 4, 6, 4, 6\}$. The cycle repeats so that term $n$ is the same as term $n+2$. The tenth number in the cycle is $6$, giving an answer of $\boxed{D}$.

Solution 2

Again, the value we seek is equal to $(2\cdot 4\cdot 6\cdot 8)^{10}\mod 10$. We can use equivalence to simplify.

$(2\cdot 4\cdot 6\cdot 8)^{10}\mod 10$

$\equiv (2\cdot 4\cdot (-4)\cdot (-2))^{10} \mod 10$

$\equiv 64^{10} \mod 10$

$\equiv 4^{10} \mod 10$

$\equiv 2^{20} \mod 10$

$\equiv 8^6\cdot 2^2 \mod 10$

$\equiv (-2)^6\cdot 2^2 \mod 10$

$\equiv 2^8 \mod 10$

$\equiv 256 \mod 10$

$\equiv 6 \mod 10$

Thus our answer is $\boxed{D}$.

See Also

1997 AJHSME (Problems • Answer Key • Resources)
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