Question

Without using the Fundamental Theorem of Arithmetic, use strong induction to prove that for all positive integers n with n ≥ 2, n has a prime factor.

Answer #1

1. The Fundamental Theorem of Arithmetic states: Every integer
greater than or equal to 2 has a unique factorization into prime
integers. Prove by induction the uniqueness part of the Fundamental
Theorem of Arithmetic.

Using induction prove that for all positive integers n, n^2−n is
even.

use the fundamental theorem of arithmetic to prove:
if a divides bc and gcd(a,b)=1 then a divides c.

Use the Strong Principle of Mathematical Induction to prove that
for each integer n ≥28, there are nonnegative integers x and y such
that n= 5x+ 8y

Use strong induction to prove that every natural number n ≥ 2
can be written as n = 2x + 3y, where x and y are integers greater
than or equal to 0. Show the induction step and hypothesis along
with any cases

Prove every integer n ≥ 2 has a prime factor. (You cannot just
cite the Funda- mental Theorem of Arithmetic; this was the first
step in proving the Fundamental Theorem of Arithmetic

Prove by induction that 5n + 12n – 1 is divisible by 16 for all
positive integers n.

Prove by induction that 5^n + 12n – 1 is divisible by 16 for all
positive integers n.

Prove that a positive integer n, n > 1, is a perfect square
if and only if when we write
n =
P1e1P2e2...
Prer
with each Pi prime and p1 < ... <
pr, every exponent ei is even. (Hint: use the
Fundamental Theorem of Arithmetic!)

5. Use strong induction to prove that for every integer n ≥ 6,
we have n = 3a + 4b for some nonnegative integers a and b.

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