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Prove If d Divides a and b Then d Squared Divides ab | Prime and Composite Numbers

by |Published
The Ultimate Crash Course for STEM Majors bundle

Sample from The Ultimate Crash Course Series

This lesson is a sample from the Ultimate Crash Course bundle. Get the full collection at Payhip and access over 1,000 lessons and podcasts through theSTEMmajor.com.

Prove If d \mid a and d \mid b, Then d^2 \mid ab

This number theory lesson shows how to prove a divisibility statement using the definition of divisibility, and then reviews the meanings of prime numbers, composite numbers, and even prime examples.

Question

Prove or disprove the following statement. Let a, b, and d be integers with d \ne 0. If d \mid a and d \mid b, then d^2 \mid ab.

Proof

Assume d \in \mathbb{Z} \setminus \{0\}.

If d \mid a and d \mid b, then by the definition of divisibility there exist integers q,r \in \mathbb{Z} such that

a=dq \quad \text{and} \quad b=dr.

Now multiply the two expressions:

ab=(dq)(dr)=d^2(qr).

Since integers are closed under multiplication, qr \in \mathbb{Z}. Therefore ab can be written as d^2 times an integer. By the definition of divisibility, this means

d^2 \mid ab.

Thus, the statement is true.

Q.E.D.

Section 2: Unique Factorization

Reference: Dudley Underwood, Elementary Number Theory, 2nd ed. New York: W. H. Freeman and Company, 1989.

Definitions

Prime Number

  • An integer greater than 1 that has no positive divisors other than 1 and itself.

Composite Number

  • An integer greater than 1 that is not prime.

Example 1: How Many Even Primes Are There

How many even primes are there? How many prime numbers have a last digit of 5?

An even number greater than 1 can be written as

n=2k, \quad k \in \mathbb{N}.

An odd number greater than 1 can be written as

n=2k+1, \quad k \in \mathbb{N}.

Part 1: How Many Even Prime Numbers Are There

Since a prime number is defined to be an integer greater than 1, the first prime number is 2. Every even number greater than 2 is divisible by 2, so it has at least three positive factors: 1, 2, and itself. Therefore, 2 is the only even prime number.

The only even prime number is 2.

Part 2: How Many Prime Numbers End in 5

Any integer ending in 5 is divisible by 5. Therefore, any number greater than 5 that ends in 5 cannot be prime. The only prime number whose last digit is 5 is 5 itself.

The only prime number that ends in 5 is 5.

Why This Number Theory Example Matters

These ideas form part of the foundation of number theory. Understanding divisibility, prime numbers, composite numbers, and basic proof writing is essential for students studying abstract algebra, discrete mathematics, computer science, and higher mathematics.

Sample from The Ultimate Crash Course Series

This is just a sample from the Ultimate Crash Course Series. To access the full bundle of lessons, worked examples, proof practice, and guided math explanations, visit Payhip. Students can also access over 1,000 additional lessons and podcasts at theSTEMmajor.com.

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