Name: Composite Number as Product of Primes
Uploaded: 2019-07-19
Duration: 6 min 23 s
Description: How do you write number 81? Number 81 can be written as 3x3x3x3, which cannot be divided further. What is this called? Explore more about it in this video.

Question

Use Euclid’s division lemma to show that the square of any positive integer is either of the form

3 m

or

3 m + 1

for some integer

m

.

Accepted Answer

Let $x$ be any positive integer.

We need to prove that $x^{2}$ is in the form of $3 m$ or $3 m + 1$ where $m$ is an integer.

Let $b = 3$ be the other integer, hence applying Euclid's division lemma to $x$ and $b = 3$ .

Then, $x = 3 q + r$ for another integer $q \geq 0$ and $r = 0, 1, 2$ because $0 \leq r < 3$ .

Therefore, $x = 3 q$ , for $r = 0$ or $3 q + 1$ , for $r = 1$ or $3 q + 2$ , for $r = 2$ .

Now consider $x^{2}$

$x^{2} = {(3 q)}^{2}$ or ${(3 q + 1)}^{2}$ or ${(3 q + 2)}^{2}$

$x^{2} = (9 q^{2})$ or $9 q^{2} + 6 q + 1$ or $9 q^{2} + 12 q + 4$

$x^{2} = 3 \times (3 q^{2})$ or $3 (3 q^{2} + 2 q) + 1$ or $3 (3 q^{2} + 4 q + 1) + 1$

$x^{2} = 3 k_{1}$ or $3 k_{2} + 1$ or $3 k_{3} + 1$

where $k_{1} = 3 q^{2}$ , $k_{2} = 3 q^{2} + 2 q$ and $k_{3} = 3 q^{2} + 4 q + 1$ since $q, 2, 3, 1$ etc are all integers, so their sum and product will be an integer.

So, $k_{1,} k_{2,} k_{3}$ are all integers.

Hence, it can be said that the square of any positive integer is either of the form of $3 m$ or $3 m + 1$ for any integer $m$ .

Use Euclid’s division lemma to show that the square of any positive integer is either of the form $3 m$ or $3 m + 1$ for some integer $m$ .

Important Questions on Real Numbers

Learn and Prepare for any exam you want !

Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m+1 for some integer m.

Important Questions on Real Numbers

Learn and Prepare for any exam you want !

Use Euclid’s division lemma to show that the square of any positive integer is either of the form $3 m$ or $3 m + 1$ for some integer $m$ .