Download 1. Prove that 3n + 2 and 5 n + 3 are relatively prime for every positive

1. Prove that 3n + 2 and 5n + 3 are relatively prime for every positive integer n .
2. Find d = gcd(574, 252) and solve the equation
572a + 252b = 28
with a, b ∈ Z
3. Show that if n ≥ k ≥ r ≥ 0 we have
! !
!
!
n k
n n−r
=
k
r
r
k−r
!
n
X n
4. Prove that
= 2n , for every n ≥ 1.
k
k=0
5. Use the Euclidean algorithm to compute the greatest common divisor of 1197 and
14280, and to express it as a linear combination of 1197 and 14280.
6. Show that for n ≥ 1:
!
2n
1 · 3 · 5 · · · (2n − 1)
× 22n
=
n
2 · 4 · 6 · · · 2n
7. Prove that 3a2 − 1, a ∈ Z can never be a perfect square, i.e = b2 , for some integer
number b.
8. Show that gcd(2a + 1, 9a + 4) = 1 , ∀a ∈ Z.
9. Define a sequence of integers as follows:
f0 = 0,
f1 = 1,
fn = fn−1 + fn−2 ,
n≥2
Prove that gcd(fn , fn+1 ) = 1 for every n ≥ 1.
10. If x is any real number, define bxc = the greatest integer less than or equal to x
(the floor of x) and dxe the least integer greater than or equal to x (the ceiling of x).
Define also {x} = x − bxc the fractional part of x.
• What are b−1.1c, b0.99999c, d−1.1e, d1.01e?
• Prove that for every x ∈ R b−xc = −dxe.
• Whichp of the following equations are true for all positive real numbers x?
√
a) bp bxcc = b xc
√
b) d bxce = d xe
11. Factor 51, 948 into a product of primes.
12. Prove that n5 − n is divisble by 30 for every integer n.
13. Prove that n, n + 2, n + 4 are all primes if and only if n = 3.
n
14. The integer Fn = 22 +1 is called the nth Fermat number. Show that for n = 1, 2, 3, 4
Fn is prime.
15. Show that F5 is not prime.
16. Prove that the equation
3x1 + 5x2 = b
has a solution in non-negative integers for b = 0, 3, 5, 6 and all b ≥ 8.
17. Prove a3 ≡ a ( mod 6) for every integer a.
18. Prove that a4 ≡ 1 ( mod 5) for every integer a that is not divisble by 5.