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\n" ); document.write( "(1) The sum of n consecutive integers is n, multiplied by the average of the first and last terms.
\n" ); document.write( "(2) If n is odd, then the first and last terms are either both even or both odd; that means the sum of the first and last terms is even, so the average of the first and last terms is an integer.
\n" ); document.write( "(3) If n is even then the first and last terms are one even and one odd; that means the sum of the first and last terms is odd, so the average of the first and last terms is halfway between two integers.
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\n" ); document.write( "(a) Find the smallest positive integer which can be written as the sum of three, four, five or seven consecutive integers.
\n" ); document.write( "Item (2) above tells us that the sum of 3 consecutive integers is divisible by 3, the sum of 5 consecutive integers is divisible by 5, and the sum of 7 consecutive integers is divisible by 7. So a number that can be written as a sum of 3 or 5 or 7 consecutive integers has to be divisible by 3 and by 5 and by 7; that means it has to be divisible by the LCM of 3, 5, and 7, which is 3*5*7 = 105.
\n" ); document.write( "So 105, and any positive integer multiple of 105, can be written as the sum of 3, 5, or 7 consecutive integers.
\n" ); document.write( "So the positive integers that can be written as the sum of 3, 5, or 7 consecutive integers are of the form 105k, where k is a positive integer.
\n" ); document.write( "We are looking for the smallest positive integer that can be written as the sum of 3, or 4, or 5, or 7 consecutive integers. We know that any number of the form 105k can be written as the sum of 3 or 5 or 7 consecutive integers; what additional requirement do we need to make a number of that form also able to be written as the sum of 4 consecutive integers?
\n" ); document.write( "Item (3) above tells us that the average of n consecutive integers is halfway between two integers if n is even.
\n" ); document.write( "If the number 105k is to be written as the sum of 4 consecutive integers, then item (3) above tells us that 105k divided by 4 must be halfway between two integers. That is going to be true only if k is even.
\n" ); document.write( "So the positive integers that can be written as the sum of 3, or 4, or 5, or 7 consecutive integers are of the form 105(2k), or 210k.
\n" ); document.write( "Then of course the smallest of those is 210.
\n" ); document.write( "ANSWER, part (a): 210
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\n" ); document.write( "(b) Explain why it is not possible for a number to be written as both the sum of four consecutive integers and the sum of six consecutive integers.
\n" ); document.write( "Item (3) above tells us that the sum of 4 consecutive integers is 4 times a number halfway between two integers: 4(a+1/2) = 4a+2. That sum is always even.
\n" ); document.write( "Item (3) above also tells us that the sum of 6 consecutive integers is 6 times a number halfway between two integers: 6(b+1/2) = 6b+3. That sum is always odd.
\n" ); document.write( "So the sum of 4 consecutive integers is always even, while the sum of 6 consecutive integers is always odd. So clearly there are no numbers that can be written both as the sum of 4 consecutive integers and as the sum of 6 consecutive integers.
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