document.write( "Question 60968: find the units digit of 3^13*7^14 without actual multiplication \n" ); document.write( "

Algebra.Com's Answer #41908 by joyofmath(189) $\"\"$ $\"About$

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We're looking at $\"3%5E13+%2A+7%5E14\"$ \r
\n" ); document.write( "\n" ); document.write( "Well, $\"3%5E1+=+3\"$ so $\"3%5E1\"$ ends in 3.
\n" ); document.write( " $\"3%5E2\"$ ends in 3*3 or 9.
\n" ); document.write( " $\"3%5E3\"$ ends in 9*3 =27 so it ends in 7.
\n" ); document.write( " $\"3%5E4\"$ ends in 7*3 = 21 so it ends in 1.
\n" ); document.write( " $\"3%5E5\"$ ends in 1*3 or 3. So it ends in the same as $\"3%5E1\"$ .
\n" ); document.write( " $\"3%5E6\"$ ends in 3*3 ends the same as $\"3%5E2\"$ .
\n" ); document.write( " $\"3%5E7\"$ ends in 9*3 ends the same as $\"3%5E3\"$ .
\n" ); document.write( "So, the pattern repeats every 4 powers. So, $\"3%5E13\"$ ands with the same digit as $\"3%5E9\"$ and $\"3%5E5\"$ which is 3.\r
\n" ); document.write( "\n" ); document.write( " $\"7%5E1\"$ ends in 7.
\n" ); document.write( " $\"7%5E2\"$ ends in the same digit as 7*7, or 9.
\n" ); document.write( " $\"7%5E3\"$ ends in the same digit as 9*7, or 3.
\n" ); document.write( " $\"7%5E4\"$ ends in the same digit as 3*7, or 1.
\n" ); document.write( " $\"7%5E5\"$ ends in the same digit as 1*7, which ends the same way as $\"7%5E1\"$ .
\n" ); document.write( "So, this pattern also repeats every 4 powers.
\n" ); document.write( "So, $\"7%5E14\"$ ends the same way as $\"7%5E10\"$ and $\"7%5E6\"$ and $\"7%5E2\"$ which is with a 9.\r
\n" ); document.write( "\n" ); document.write( "So, $\"3%5E13+%2A+7%5E14\"$ ends with the same digit as $\"3%5E13\"$ which ends in 3 multiplied by the last digit of $\"7%5E14\"$ which is 9. \r
\n" ); document.write( "\n" ); document.write( "So, the big product ends in $\"3%2A9+=+27\"$ so 7 is the last digit of the big product. \n" ); document.write( "

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