2, Name:
Midterm Math 245 - Fall 2010
Duration: 50 minutes
To get full credit you should explain your answers.
Each of the rst three questions is worth 5 points.
The bonus questions is worth 3 points.
1. Find all the integer solutions (x, y ) of the equation
19x + 71y = 5.
#
Score
1
2
3
4
Total
2. How many natural numbers with x x 500 have no divisor in {2, 3, 5} ?
3. State and give a short proof of the binomial (
formula.) Use the binomial formula to
2 20
25
determine the coecient of x in the expansion of x2 x3 .
4. Bonus Question! Prove that every set of 8 distinct integers contains a pair whose sum
or dierence is divisible by 12.