Funny Square

Algebra Level 1

Without manually calculating find the square of the number 995

909925 900925 990025 990925

4 solutions

Lakshminarayana Potukuchi
Jun 4, 2015

995 x 995 = 99 x 100 + append 25

Moderator note:

Why does this work? Can you explain the reasoning behind this?

Dear Challenge Master and @Lakshiminarayana Potukuchi ,

Let a number be a5 where a can be any integer. The value of the number is therefore $10a+5$ . We have $(10a+5)^2 =100a^2 + 2(10a)(5) + 25$

$= 100a^2 +100a + 25 = 100a(a+1) + 25$ .

Evidently, $100a(a+1)$ ends with two zeroes so adding it to $25$ , the last two digits are always $25$ . As for the digits preceding $25$ , they are only determined by the value of $a(a+1)$ so we have a(a+1)25 as the square. Hope this helps.

Noel Lo - 6 years ago

What do you mean by append?

Omkar Kulkarni - 6 years ago

to find the square of any number that ends with 5; simply remove 5 from it and multiply the number with 1 added to it; then append 25 to it. Eg. 35 x 35 = (3 4) + append 25 i.e. 1225 Eg. 45 x 45 = (4 5) + append 25 i.e. 2025

Lakshminarayana Potukuchi - 6 years ago

Oh okay. Thank you! :)

Omkar Kulkarni - 6 years ago

Sai Teja Prasanth
Jun 4, 2015

995^2=(1000-5)(1000-5)=1000000+25-10000 by the principle of (a-b)^2=a^2+b^2-2ab

WeiLiang Wu
Jun 9, 2015

1000^2 - 995^2 = (1000 + 995) (1000 - 995)

1000^2 - 995^2 = (1995) (5)

1000000 - (1995) (5) = 995^2

1000000 - 9975 = 995^2 = 990025

Jaka Ong
Jun 7, 2015

(995+5)(995-5)+25=990025

Funny Square

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