document.write( "Question 1140609: a list of whole numbers from 1 to 50 is written on a piece of paper. all the multiples of 5 are then struck off from the list. what is the last digit of the product of the remaining numbers. \n" ); document.write( "
Algebra.Com's Answer #761136 by Edwin McCravy(20063)\"\" \"About 
You can put this solution on YOUR website!
a list of whole numbers from 1 to 50 is written on a piece of paper. all the
\n" ); document.write( "multiples of 5 are then struck off from the list. what is the last digit of the
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document.write( "There is an easy way and a hard way:\r\n" );
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document.write( "Here's the easy way:\r\n" );
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document.write( "The last digits of the whole numbers from 1 to 50 (inclusive) make up 5 groups\r\n" );
document.write( "of 1,2,3,4,5,6,7,8,9,0.  When we take away the multiples of 5, we take away\r\n" );
document.write( "those ending in 0 or 5, and that leaves 5 groups of 1+2+3+4+6+7+8+9 = 40, and\r\n" );
document.write( "adding any whole number of groups of 40 will end up having a 0 for the last\r\n" );
document.write( "digit of the sum.\r\n" );
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document.write( "Answer: 0\r\n" );
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document.write( "Now for the hard way:\r\n" );
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document.write( "First we find the sum of 1+2+3+···+48+49+50\r\n" );
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document.write( "The formula for the sum of the first N whole numbers, since they form an\r\n" );
document.write( "arithmetic sequence where a1 = 1, n=50, and an = a50 = 50 \r\n" );
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document.write( "\"S%5Bn%5D=expr%28n%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29\" becomes\r\n" );
document.write( "\"S%5B50%5D=expr%2850%2F2%29%281%2B50%29\"\r\n" );
document.write( "\"S%5B50%5D=25%2851%29\"\r\n" );
document.write( "\"S%5B50%5D=1275\"\r\n" );
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document.write( "Now from that we must subtract this sum: 5+10+15+···+40+45+50, which forms \r\n" );
document.write( "an arithmetic sequence where a1 = 5. But how am I going to find out\r\n" );
document.write( "what to substitute for n? This way:\r\n" );
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document.write( "If we divide every number in this series\r\n" );
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document.write( "5+10+15+···+40+45+50\r\n" );
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document.write( "by 5, we get this series:\r\n" );
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document.write( "1+2+3,···,8+9+10\r\n" );
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document.write( "and since we know that has 10 terms, so does the first one!  So n=10,\r\n" );
document.write( "and an = a10 = 50,  \r\n" );
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document.write( "\"S%5Bn%5D=expr%28n%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29\" becomes\r\n" );
document.write( "\"S%5B10%5D=expr%2810%2F2%29%285%2B50%29\"\r\n" );
document.write( "\"S%5B10%5D=5%2855%29\"\r\n" );
document.write( "\"S%5B10%5D=275\"\r\n" );
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document.write( "So to get the final answer the hard way, we subtract:\r\n" );
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document.write( "1275-275 = 1000, which ends in 0.   <-- final answer\r\n" );
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document.write( "The easy way is better! J\r\n" );
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document.write( "I confess! I did it the hard way first, and then the easy way dawned on me and I felt stupid!\r\n" );
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document.write( "Edwin

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