Question

Use division algorithm to show that the square of any positive integer is of the form 5m or
5 +1 or 5m + 4 where m is a whole number​

Answer 1

Given:

A positive integer of the form 5m+r.

To Show:

square of any positive integer is of the form 5m or  5m +1 or 5m + 4 where m is a whole number​.

Solution:

Any number can be represented by the form 5m+r ,

where r can be 0,1,2,3,4 and m ∈ N

Let Q be a positive integer.

By Euclid's Division Lemma,

• Q = 5m + r

Squaring Q,

• Q² = (5m + r)² = 25m² + 10mr + r²
• Q² = 5 ( 5m² + 2mr) + r²

We can take 5m² + 2mr as a number K.

Then Q² becomes,

• Q² = 5K + r².

Since r ∈ { 0, 1, 2, 3, 4}

• r² ∈ {0,1,4,9,16}

• We also have condition that any number of the form aq + r , r ≤ a , since r is the remainder when the number is divided by a.

• Therefore r² < 5
• Possible values of r² = { 0 , 1, 4 }

Therefore any positive integer is of the form, 5m , 5m +1 or 5m+4.

Answer 2

Answer:

Solution:

Any number can be represented by the form 5m+r ,

where r can be 0,1,2,3,4 and m ∈ N

Let Q be a positive integer.

By Euclid's Division Lemma,

Q = 5m + r

Squaring Q,

Q² = (5m + r)² = 25m² + 10mr + r²

Q² = 5 ( 5m² + 2mr) + r²

We can take 5m² + 2mr as a number K.

Then Q² becomes,

Q² = 5K + r².

Since r ∈ { 0, 1, 2, 3, 4}

r² ∈ {0,1,4,9,16}

We also have condition that any number of the form aq + r , r ≤ a , since r is the remainder when the number is divided by a.

Therefore r² < 5

Possible values of r² = { 0 , 1, 4 }

Therefore any positive integer is of the form, 5m , 5m +1 or 5m+4.

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