document.write( "Question 833337: The cube of 99 is 970299. Consider the smallest two-digit number whose cube ends with the original two-digit number. The sum of the digits of the cube of that two-digit number is:\r
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\n" ); document.write( "\n" ); document.write( "a) 18 b) 19 c)20 d)21 e)22 \n" ); document.write( "

Algebra.Com's Answer #502557 by KMST(5328) $\"\"$ $\"About$

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Twelve-year old geniuses at the forum in the artofproblemsolving website may be able to provide a brilliant and concise explanation showing why choices b, c, d, and e are wrong. My explanation will not be so brilliant.
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\n" ); document.write( "One strategy would be using a spreadsheet to have the computer calculate $\"n%5E3-n\"$ and see if it ends in 00 for all numbers $\"n\"$ between 11 and 99.
\n" ); document.write( "Without using a computer, I need a different strategy.
\n" ); document.write( "Algebra tells me that
\n" ); document.write( " $\"n%5E3-n=n%28n%5E2-1%29=n%28n%2B1%29%28n-1%29\"$ .
\n" ); document.write( "Since $\"%28n-1%29\"$ , $\"n\"$ and $\"%28n%2B1%29\"$ are 3 consecutive numbers, all I need to do is look for 3 consecutive integers between 10 and 100 whose product ends in 00.
\n" ); document.write( "If the product ends in 00, it must be a multiple of $\"100=4%2A25=2%5E2%2A5%5E2\"$ .
\n" ); document.write( "Because the product is a multiple of $\"4=2%5E2\"$ , one of the numbers must be a multiple of $\"4\"$ , or else two of the numbers ( $\"%28n-1%29\"$ , and $\"%28n%2B1%29\"$ ) must be even (multiples of $\"2\"$ ).
\n" ); document.write( "Also, one of the 3 numbers must be a multiple of $\"25\"$ , because we could not have two multiples of $\"5\"$ within the 3 consecutive integers.
\n" ); document.write( "So the products must include $\"25\"$ , or $\"50\"$ , or $\"75\"$ or $\"100\"$ .
\n" ); document.write( "For the smallest $\"n\"$ , I should make $\"25\"$ one of the 3 integers.
\n" ); document.write( "The 3 consecutive integers including $\"25\"$ would be smallest if we make $\"n%2B1=25\"$ . In that case, for $\"n%28n%2B1%29%28n-1%29\"$ to be a multiple of $\"100\"$ , we would need $\"n\"$ (the only even integer of the three) to be a multiple of 4.
\n" ); document.write( "Since $\"n%2B1=25\"$ means $\"n=24=4%2A6\"$ , $\"highlight%2824%29\"$ is the smallest two-digit number whose cube ends with the original two-digit number.
\n" ); document.write( "Since $\"n=24=3%2A8\"$ , $\"n%5E3\"$ is a multiple of 9, and the sum of its digits must be a multiple of 9. It could not be 19, or 20, or 21, or 22.
\n" ); document.write( "If it is one of the options given, it must be $\"highlight%28%22a+%29+18%22%29\"$ . \n" ); document.write( "

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