document.write( "Question 670006: Find the units digit of 7^157. \n" ); document.write( "

Algebra.Com's Answer #416713 by MathLover1(20850) $\"\"$ $\"About$

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To solve a question like this, one looks at $\"patterns\"$ of powers and a $\"cycle\"$ of $\"length\"$ . \r
\n" ); document.write( "\n" ); document.write( " $\"7%5E1+=7\"$ .................unit's dig. $\"7\"$
\n" ); document.write( " $\"7%5E2+=+49\"$ ...............unit's dig. $\"9\"$
\n" ); document.write( " $\"7%5E3+=+343\"$ ..............unit's dig. $\"3\"$
\n" ); document.write( " $\"7%5E4+=+2401\"$ .............unit's dig. $\"1\"$ ...pattern: $\"7\"$ , $\"9\"$ , $\"3\"$ , $\"1\"$
\n" ); document.write( " $\"7%5E5+=+16807\"$ ............
\n" ); document.write( " $\"7%5E6+=+117649\"$ ...........
\n" ); document.write( " $\"7%5E7+=+823543\"$ ...........
\n" ); document.write( " $\"7%5E8+=+5764801\"$ ...........pattern is repeating
\n" ); document.write( " $\"7%5E9+=+40353607\"$ ......
\n" ); document.write( " $\"7%5E10+=282475249\"$ .....
\n" ); document.write( " $\"7%5E11+=1977326743\"$ .....
\n" ); document.write( " $\"7%5E12+=13841287201\"$ ......pattern is repeating\r
\n" ); document.write( "\n" ); document.write( "So the unit's digits of powers of $\"7\"$ follow a cycle of length $\"4\"$ : $\"7\"$ , $\"9\"$ , $\"3\"$ , $\"1\"$ ... ... \r
\n" ); document.write( "\n" ); document.write( "so let's find the remainder when we divide $\"157\"$ by $\"4\"$ : $\"157+=+39%2A4+%2B+1+\"$
\n" ); document.write( "Since the remainder is $\"1\"$ , and $\"7%5E157\"$ has the same unit's digit as $\"7%5E1\"$ , which is $\"7\"$ .\r
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