Prove no square number ends in an 8

[Stedman]

Right im currently studying for my A level mathes at home and im trying to prove that no square number ends in an 8.

Thing is, im having no luck with it.

Anyone care to explain?

Cheers.

[10 Pence Short]

Ask KatieK, she's good with maths.

[cazzer]

Write a program that prints out the first 1 million square numbers.
I'd class thats a proof enough

[Geoff82]

I'm not sure off the top of my head, but if it were me, I'd be starting with (x^2+8) where x^2 = any integer and work back from there.

My A level maths is VERY rusty though...

[scorp]

cazzer said:

Write a program that prints out the first 1 million square numbers.
I'd class thats a proof enough

I was bored and did just that.. no number ended in 8

[ih8thisname]

x^2 Anything ending in an 8

[jon.nic]

If you write down the first ten numbers, and their squares, you see that they end in:
1, 4, 9, 6, 5, 6, 9, 4, 1, 0.

Carry on, and they carry on with that pattern (11 squared is 121, 21 squared is 441, 31 squared is 961...)

So, square numbers can only ever end with one of these numbers (the 'proof' bit is that the 'units' of the square is always the product of two numbers under 10)

That may help.....

Jon

[Geoff82]

Something is coming back to me: Have you tried using proof by induction?

Unfortunately all my best work is done by 12:00, its all downhill from lunchtime onwards.

ETA: jon.nic is inferring the same thing. If you prove it for P(n) = k and also for P(n) = k + 1 then it must be true for every integer value of n.

Edited by Geoff82 on Tuesday 10th February 18:34

[s2art]

Just do it exhausively from 1 to 9. Only the last digit makes a difference to if a number ends in 8.

[mybrainhurts]

Try asking here

[scorp]

s2art said:

Just do it exhausively from 1 to 9. Only the last digit makes a difference to if a number ends in 8.

makes sense to me.

[KatieK]

Any number can be written like

n = 10a + b

where b is between 0 and 9

So if you square it you get

n2 = 100a2 + 20ab + b2 where 2 is square symbol

so n2 has the same last digit as b2. If you square 0 to 9 you can see the last digit in the square of a number must end in 0,1,4,5,6 or 9

[Snake the Sniper]

I don't think there is a pure maths proof to this, more a case of showing it isn't so. As said, it's probably easier to show that the final digits follow a recurring pattern and go from there.

[Famous Graham]

Shouldn't "number" = "integer" in all the instances quoted?

Or am I misremembering the definitions?

[cazzer]

scorp said:

cazzer said:

Write a program that prints out the first 1 million square numbers.
I'd class thats a proof enough

I was bored and did just that.. no number ended in 8

Sorted.....just say Scorp proved it.....

[DrTre]

cazzer said:

scorp said:

cazzer said:

Write a program that prints out the first 1 million square numbers.
I'd class thats a proof enough

I was bored and did just that.. no number ended in 8

Sorted.....just say Scorp proved it.....

Aye, just link to this thread. Bish bosh.

[ian in lancs]

10 Pence Short said:

Ask KatieK, she's good with maths.

Where's your answer then smartarse?

[10 Pence Short]

ian in lancs said:

10 Pence Short said:

Ask KatieK, she's good with maths.

Where's your answer then smartarse?

I didn't say I was good with maths. That's why I sugested someone else.

[deckster]

KatieK said:

Any number can be written like

n = 10a + b

where b is between 0 and 9

So if you square it you get

n2 = 100a2 + 20ab + b2 where 2 is square symbol

so n2 has the same last digit as b2. If you square 0 to 9 you can see the last digit in the square of a number must end in 0,1,4,5,6 or 9

Egad, I believe she's got it! KatieK in 'not posting ste' shocker!

[10 Pence Short]

scorp said:

10 Pence Short said:

Ask KatieK, she's good with maths.

KatieK said:

Any number can be written like

n = 10a + b

where b is between 0 and 9

So if you square it you get

n2 = 100a2 + 20ab + b2 where 2 is square symbol

so n2 has the same last digit as b2. If you square 0 to 9 you can see the last digit in the square of a number must end in 0,1,4,5,6 or 9

owned.

Absolutely, all I can say is well done to KatieK on that one.

I'm off to spend a few hours self flagellating for my insolence.