*Bite-size chunks of incredibly important mathematics*

*Bite-size chunks of incredibly important mathematics*

**Proof**

**Proof**

**Some basics:**

If *n* is an integer,* *consecutive integers could be either side i.e. *n-1, n, n+1, n+2 *etc.

Regardless of whether *n* is even or odd, 2*n* will be even, and *2n-1*, and *2**n+1* will be odd

Multiples of 2, 3 and 5 are written 2*n, 3n, 5n* respectively

##### “Prove algebraically that the sum of two even numbers is even”

Let

mandnbe two numbers, then 2m and2nwill be even numbers, so

As this is a multiple of 2, it is an even number

##### “Prove algebraically that the sum of two odd numbers is even”

Let

andmbe two numbers, thenn2m+1and2n+1will be odd numbers, so

As this is a multiple of 2, it is an even number

##### “Prove algebraically that the sum of an odd and an even number is odd”

Let

andmbe two numbers, thenn2m+1will be odd and2nwill be even, so

is a multiple of 2, so it is an even number

Therefore, is an odd number

##### “Prove algebraically that the sum of three consecutive integers is divisible by three”

Let

nbe an integer, thenn+1, andn+2will be consecutive integers.

Since 3 is a factor of this result, so the sum of the 3 consecutive integers will be divisible by 3.

##### “Prove that the product of any three consecutive integers is divisible by 6”

In any 3 consecutive integers, one number is always a multiple of 2, and one will be a multiple of 3

Let the multiple of 2 be written 2n and the multiple of 3 be written 3m

Their product is (2n)(3m) = 6mn

Therefore, the product having a factor of 6, is divisible by 6.

##### “Prove algebraically that the difference between the squares of any two odd integers is divisible by 8”

Let two integer numbers be

mandn, then2m+1and2n+1will both be odd numbersUsing the difference of two squares i.e.

So 4 is a factor

If m and n are both even then (m-n) is even, and has a factor of 2

If m or n is even and the other odd then (m+n+1) is even and has a factor of 2

Therefore is divisible by 8

##### “Prove algebraically that the difference between the squares of any two consecutive integers is equal to the sum of the two integers”

Let

nbe an integer, thenn+1will be a consecutive integer.

This represents the sum of the two consecutive integers.

##### “Prove algebraically that the sum of the squares of any two consecutive numbers always leaves a remainder of 1 when divided by 4”

Let

nbe an integer, thenn+1will be a consecutive integer.

has a factor 2, and as

nandn+1are consecutive, one must be even, and therefore be a multiple of 2So must be divisible by 4

Therefore must have remainder 1 when divided by 4

##### “Prove algebraically that the sum of two consecutive multiples of 5 is always an odd number”

Let

nbe an integer, then5nis a multiple of 5, and5(n+1)will consecutively be the next multiple of 5

Whether

nis even or odd, will be oddSince 5 and are odd numbers, their product is an odd number

##### “Prove algebraically that the product of two consecutive multiples of 5 is always an even number”

Let

nbe an integer, then5nis a multiple of 5, and5(n+1)will consecutively be the next multiple of 5

Either

nwill be odd andn+1will be even, or vice versaSince 25 is odd, odd x odd x even = even, i.e. their product is an even number