Don't Use The Calculator

Number Theory Level 3

Given that $2^{29}$ is a nine digit number with all the digits distinct. Which is the missing digit?

2 solutions

Ansh Bhatt
May 7, 2015

The remainder obtained on dividing a number by nine is equal to the remainder obtained on dividing the sum of the digits of that number by nine.

Remainder on dividing (2^29) by 9 = 5

taking the missing digit as 'x'

(0 + 1 + 2 +..... + 9 - x) when divided by 9 gives remainder 5

.'. 45 - x - 5 = 9k

.'. 40 - x = 9k

x is a single digit number

.'. x = 4

This is a clever solution, Ansh. It took me a moment to realize that

$2^{3} \equiv -1 \pmod{9} \Longrightarrow 2^{29} = (2^{3})^{9}*2^{2} \equiv -4 \pmod{9} \equiv 5 \pmod{9}.$

Brian Charlesworth - 6 years, 1 month ago

thanx, @brian i am not at all good at formatting the text so i was at a loss on how to explain the method. you did it, so thanx.

Ansh Bhatt - 6 years, 1 month ago

Good question and a great solution!

Sravanth C. - 6 years, 1 month ago

how do u divide 2^29 by 9 without a calculator?Otherwise, Gr8 solution!

Mayukh Biswas - 6 years, 1 month ago

Using modular arithmetic, as I did in my comment below. :)

Brian Charlesworth - 6 years, 1 month ago

Well done.

Raghav Vaidyanathan - 6 years, 1 month ago

Senthil Kumar
May 15, 2015

Beautiful and simple problem using modular arithmetic.