document.write( "Question 1084381: If a(k) = 5+5^2+5^3+5^4+...+5^k , for which of the following values of k will a(k) be divisible by 10?
\n" ); document.write( "F) 37 G) 51 H) 75 J) 88 K) 91 \n" ); document.write( "

Algebra.Com's Answer #698480 by KMST(5328) $\"\"$ $\"About$

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$\"highlight%28%22J+%29+88%22%29\"$
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\n" ); document.write( "THE FAST WAY TO THE ANSWER:
\n" ); document.write( "All the terms are odd numbers, and you can never get an even sum (such as any multie of 10) by adding an odd number of terms.
\n" ); document.write( "That eliminates all the choices, except the one above.
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\n" ); document.write( "THE SHOW-YOUR-WORK EXPLANATION:
\n" ); document.write( "That is a geometric series, the sum of consecutive terms of a geometric sequence (or geometric progression, depending on where you study math).
\n" ); document.write( "The sum in this problem is the sum of the first k terms of a geometric sequemce
\n" ); document.write( "starting with first term $\"green%285%29\"$ , and with a common ratio of $\"red%285%29\"$ .
\n" ); document.write( "That is $\"green%285%29%2A%28red%285%29%5Ek-1%29%2F%28red%285%29-1%29=5%2A%28red%285%29%5Ek-1%29%2F4\"$ .
\n" ); document.write( "For that to be a multiple of 10, we need $\"red%285%29%5Ek-1%29\"$
\n" ); document.write( "to be a multiple of 8.
\n" ); document.write( "There is a way to look at numbers grouping them according to divisibity by a fixed number of interest to us.
\n" ); document.write( "All numbers divisible by 8 would be said to be congruent with zero modulo 8,
\n" ); document.write( "because they are 0 more than a multiple of 8.
\n" ); document.write( "For positive numbers, we would say they are congruent with the remainder we get when we divide them by 8.
\n" ); document.write( "(That is modular arithmetic, where you add and multiply lime a clock,
\n" ); document.write( "except for modulo 8 there would only be numbers 1 through 8 on the clock)
\n" ); document.write( "As we go from one power of 5 to the next,
\n" ); document.write( " $\"5%5E2\"$ is congruent with 1 modulo 8;
\n" ); document.write( " $\"5%5E3=5%2A5%5E2\"$ is congruent with $\"5%2A1=5\"$ ;
\n" ); document.write( " $\"5%5E4=5%2A5%5E3\"$ is congruent with $\"5%2A5=25\"$ ,
\n" ); document.write( "and also congruent with $\"1\"$ } , because $\"25\"$ is congruent with $\"1\"$ .
\n" ); document.write( " $\"5%5E5=5%2A5%5E4\"$ is congruent with $\"5%2A1=5\"$ ;
\n" ); document.write( " $\"5%5E6\"$ is congruent with $\"1\"$ ,
\n" ); document.write( " $\"5%5E7\"$ is congruent with $\"5\"$ ,
\n" ); document.write( "and so on.
\n" ); document.write( "So, $\"5%5Ek\"$ is congruent with 1 modo 8 for all even k numbers,
\n" ); document.write( "and it is congruent with 5 modulo 8 for all odd k numbers.
\n" ); document.write( "That means that $\"5%5Ek-1\"$ is congruent with $\"1-1=0\"$ modulo 8,
\n" ); document.write( "so it is a multiple of 8, making the sum a multiple of 10. \n" ); document.write( "

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