SOLUTION: A math teacher collects old books. One day, a student asked him how many old math books he has. He replied, "If I divide the books into unequal whole numbers, then 64 times the d

Algebra ->  Equations -> SOLUTION: A math teacher collects old books. One day, a student asked him how many old math books he has. He replied, "If I divide the books into unequal whole numbers, then 64 times the d      Log On


   

Question 197753: A math teacher collects old books. One day, a student asked him how many old math books he has. He replied, "If I divide the books into unequal whole numbers, then 64 times the difference between the two numbers equals the difference between the squares of the two numbers." How many old math books does the teacher have?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Call the larger number a
Call the smaller number b
given:
64%2A%28a+-+b%29+=+a%5E2+-+b%5E2
64%2A%28a+-+b%29+=+%28a+%2B+b%29%28a+-+b%29
64+=+a+%2B+b
I now know that the sum of the numbers is 64
I can think of the right side of the equation as
(hypotenuse)^2 - (one leg)^2 of a right triangle
where a is the hypotenuse and b is the leg
The left side of the equation must be c%5E2
where c+=+sqrt%2864%2A%28a+-+b%29%29
From above, b+=+64+-+a+, so
c+=+8%2Asqrt%28a+-+%2864+-+a%29%29
c+=+8%2Asqrt%282a+-+64%29
c must be a whole number, since the right side
of the original equation is a whole number
That means that sqrt%282a+-+64%29 must be a whole number
Note that if a+=+32, then c+=+0 and
c%5E2+=+0, a%5E2+=+b%5E2, a+=+b and that is
not allowed by the problem.
So a+%3E+32
I'll try a+=+33
b+=+64+-+a
b+=+64+-+33
b+=+31
c+=+8%2Asqrt%2866+-+64%29
c+=+8%2Asqrt%282%29
Not allowed, since c has to be a whole number
I'll try a+=+34
b+=+30
c+=+8%2Asqrt%2868+-+64%29
c+=+8%2Asqrt%284%29
c+=+16
c%5E2+=+256
a%5E2+-+b%5E2+=+34%5E2+-+30%5E2
a%5E2+-+b%5E2+=+1156+-+900
a%5E2+-+b%5E2+=+256
So, c%5E2+=+a%5E2+-+b%5E2
64%2A%28a+-+b%29+=+a%5E2+-+b%5E2
64%2A%2834+-+30%29+=+256
64%2A4+=+256
256+=+256
---------------
a+%2B+b+=+64
The teacher had 64 old math books