The Calculator Problem

[Gouri Basak]

Take all the 9 numbers 1,2,3,....,9 in a calculator and make three 3 - digit numbers out of it. Add those numbers . Prove that the number is divisible by 9 .

[Nitin Prasad]

Let (abc) denote the number 100a+10b+c where a,b,c are digits

Now let (abc), (def), (ghi) be the three 3-digit numbers formed using the digits from 1 to 9.

Now observe that-

$$(abc)+(def)+(ghi)

=100a+10b+c+100d+10e+f+100g+10h+i

=99a+9b+99d+9e+99g+9h+a+b+c+d+e+f+g+h+i$$

Since $$a+b+c+d+e+f+g+h+i= 1+2+3+\cdots +9=45$$

Hence (abc)+(def)+(ghi) is divisible by 9

[Author]

Posts