A number theory curiosity

Question

961 is the square of 31. If you reverse the digits of 961, you get 169 which is 13 (the reverse of 31) squared.

961 = 31^2 
 169 = reverse(961) = 13^2 = reverse(31)^2

I can only think of one other non-trivial example - 12.

Can anyone else come up with other examples? A formula for finding them? Restrictions on the size of the digits or the number of digits? What about larger powers?

Silver89 · Answer

Finding Square of a number ending with 5 
To find the square of 75
Do the following
Multiply 5 by 5 and put 25 as your right part of answer.
Multiply 7 with the next higher digit ie (7+1)=8 gives
56 as the left part of the answer, Answer is 5625

if you like similar shortcuts i can post another

mrblack · Answer

??? How did you even know how to do that? Did you just make that up?

KarishmaB · Answer

It's a very popular Vedic Math trick. Now don't rush to go and find a Vedic Math book because though fun, it is not very useful for GMAT (GMAT is not a calculation oriented test). This trick is definitely a good one and it comes in handy so worth discussing. There are other tricks involving squares as well. They are all based on the same concept.

Logic behind this trick: Consider any number ending in 5. The highlighted portion will vary but the rest will always remain the same.
 3 5^2 = ( 3 0 + 5)^2
Using  (a+b)^2 = a^2 + 2ab + b^2 , we get
= ( 3 0)^2 + 2* 3 0*5 + 5^2
=  3*3 *100 +  3 *100 + 25
=  3*4 *100 + 25
= 12 25

Now, instead let's consider (65)^2
( 6 5)^2 = ( 6 0 + 5)^2
= ( 6 0)^2 + 2* 6 0*5 + 5^2
=  6*6 *100 +  6 *100 + 25
=  6*7 *100 + 25
= 42 25

Do you see why it will end with 25 every time and why the number before the 5 gets multiplied by the next number?
People have observed these things and created shortcuts.

Similarly, there are shortcuts for squares of numbers close to 50, 100 etc. See if you can figure them out using the identity  (a+b)^2 = a^2 + b^2 + 2ab

mrblack · Answer

thanks. hats off to you.

openbox1 · Answer

Damn Karishma, you must be a quant goddess. I see your extremely clear and detailed explanations for so many qtns. A BIG THANKS from me!!!!

bumpbot · Answer

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A number theory curiosity

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