Math 312 Solutions To Practice Problems Worksheet With Answers Page 2
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1
2
32
of 2, we would have that n is divisible by some odd prime, say p. If
n = p, we have
1
2
a + 1 = a + 1 = (a + 1) a
a
+
a + 1
and so a + 1 is not prime (each factor above is > 1). If n > p,
( (
1)
(
2)
a + 1 = (a
) + 1 = (a
+ 1) a
a
+
a
+ 1
which again contradicts the fact that a + 1 is prime.
5 Show that if all three of p, p + 2 and p + 4 are prime, then p = 3.
Solution We apply the Division algorithm to p and 3 and find that
p has a remainder of either 0, 1 or 2, after division by 3. In the first
case, we necessarily have p = 3. In the second, it follows that p + 2
is divisible by 3 and hence equal to 3, contradicting the fact that p is
prime. In the third case, we have that p + 4 is divisible by and hence
equal to 3, again a contradiction.
6 Use the Euclidean algorithm to compute (2059, 2581) and to express
this quantity as a linear combination of 2059 and 2581.
Solution We have
2581 = 1 2059 + 522
2059 = 3 522 + 493
522 = 1 493 + 29
493 = 17 29 + 0
and so (2059, 2581) = 29. We thus can write
29 = 522
493 = 522
(2059
3 522) = 4 522
2059
and so
29 = 4(2581
2059)
2059 = 4 2581
5 2059.
7 Show that every nonzero integer can be uniquely expressed as
1
a 3 + a
3
+
+ a
3 + a
1
1
0
where a
1, 0, 1 and a = 0.
Solution This follows from the Division algorithm upon noting that
if, after dividing n by 3, we obtain a remainder of 2, we may write
n = 3k + 2 = 3(k + 1)
1.
8 Prove that there are infinitely many primes of the shape 6n + 5.
Solution We note that if we multiply 2 numbers of the form 6k + 1
together, we get another number of the same form (check this!). If
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