document.write( "Question 1116378: How many integers between 1 and 100 can be written as the difference of two perfect squares? \n" ); document.write( "

Algebra.Com's Answer #731276 by greenestamps(13203) $\"\"$ $\"About$

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\n" ); document.write( "We are looking to find all the numbers from 1 to 100 that can be written in the form

\n" ); document.write( " $\"a%5E2-b%5E2\"$

\n" ); document.write( "where a and b are positive integers with a>b.

\n" ); document.write( "(1) If a = b+1, then
\n" ); document.write( " $\"a%5E2-b%5E2+=+%28b%2B1%29%5E2-b%5E2+=+%28b%5E2%2B2b%2B1%29-b%5E2+=+2b%2B1\"$

\n" ); document.write( "Since b is any positive integer, this means every odd number except 1 can be written as the difference of two squares.

\n" ); document.write( "As an example, if the number is 31, then
\n" ); document.write( " $\"2b%2B1=31\"$
\n" ); document.write( " $\"2b=30\"$
\n" ); document.write( " $\"b=15\"$
\n" ); document.write( " $\"a+=+b%2B1+=+16\"$
\n" ); document.write( "and
\n" ); document.write( " $\"16%5E2-15%5E2+=+256-225+=+31\"$

\n" ); document.write( "So we know that all odd numbers except 1 can be written as the difference of perfect squares.

\n" ); document.write( "There are 50 odd numbers from 1 to 100; all but one of them can be written as the difference of perfect squares.

\n" ); document.write( "So at this point we have found 49 numbers from 1 to 100 that can be written as the difference of perfect squares.

\n" ); document.write( "(2) So what even numbers can be written as the difference of perfect squares?

\n" ); document.write( "To get an even difference of perfect squares a^2 and b^2, a and b must be either both even or both odd. In either case, the difference of squares is a multiple of 4. So we know that the only even numbers that can be written as the difference of perfect squares are those that are divisible by 4.

\n" ); document.write( "There are 50 even numbers from 1 to 100; half of them are divisible by 4, the other half are not. So we have found 25 even numbers from 1 to 100 that can NOT be written as the difference of perfect squares.

\n" ); document.write( "Finally, similar to the case for odd numbers, not ALL multiples of 4 can be written as the difference of perfect squares.

\n" ); document.write( "If a = b+2, then
\n" ); document.write( " $\"a%5E2-b%5E2+=+%28b%2B2%29%5E2-b%5E2+=+%28b%5E2%2B4b%2B4%29-b%5E2+=+4b%2B4\"$

\n" ); document.write( "Since b has to be a positive integer, the smallest multiple of 4 that CAN be written as the difference of perfect squares is 8 (when b=1).

\n" ); document.write( "So we have the answer to the question. The numbers from 1 to 100 that CAN be written as the difference of perfect squares are

\n" ); document.write( "(1) all odd numbers except 1 (49 numbers); and
\n" ); document.write( "(2) all multiples of 4 except 4 (24 numbers)

\n" ); document.write( "making a total of 73. \n" ); document.write( "

\n" );