1. Find the greatest common divisor of 1171 and 283 without factoring the numbers.
2. (a) Find the greatest common divisor of and .
(b) Suppose p and q are distinct prime numbers. Using the idea of (a), find the greatest commmon divisor of and .
3. Calvin Butterball takes an integer n and adds 1000000 to get another integer. Can both of the integers be divisible by 14? Why or why not?
If n and are both divisible by 14, then so is . Since , it follows that 14 cannot divide both n and .
4. Prove that it is not possible to find two integers m and n whose sum is 171 and whose difference is 130.
Suppose and . Adding the two equations, I obtain . Then , which is a contradiction. Thereore, it is not possible to find two integers m and n whose sum is 171 and whose difference is 130.
5. Prove that if , then and are relatively prime.
Therefore, and are relatively prime for all .
6. Prove that if , then .
and , so
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