Easy #3

Number Theory Level 2

What are the last three digits of $7777\times 9999$ ?

The answer is 223.

2 solutions

Krishna Ar
Jul 26, 2014

Since we need to find out last three digits we can use $\pmod {1000}$ . Writing this in that way gives $(-223) \times (-1) \pmod {1000}= 223$ , and we are done.

Kindly upvote my solution if you liked this simple way without having to use a calculator :)

Beautiful solution! When forbidden to use a calculator, this would certainly show how to solve the problem when more digits are asked and it would become increasingly time-consuming to calculate the part of the multiplication relevant to the digits you want to find.

Ralph Schraven - 6 years, 10 months ago

Thank you for the appreciation :)

Krishna Ar - 6 years, 10 months ago

beautifully soved dear

justine jonathan - 6 years, 9 months ago

Thank you! :)

Krishna Ar - 6 years, 9 months ago

I edited your $\LaTeX$ slightly. Note that instead of simply typing it as mod 1000, \pmod {1000} should be used. Overall, good solution :)

Victor Loh - 6 years, 10 months ago

Thanks a lot :)

Krishna Ar - 6 years, 10 months ago

Yeah no problem :)

Victor Loh - 6 years, 10 months ago

@Victor Loh – Thanks for doing these questions. Great practice for me.

Sharky Kesa - 6 years, 10 months ago

Shouldn't it be $\pmod{10000}$ ?

Daniel Lim - 6 years, 10 months ago

No because the problem requests for the last three digits.

Victor Loh - 6 years, 10 months ago

Yes, what a careless mistake by me

Daniel Lim - 6 years, 10 months ago

how do we obtain mod 1000??

Arun AR - 6 years, 10 months ago

Awesome :)

Sanjana Nedunchezian - 6 years, 9 months ago

William Isoroku
Aug 2, 2014

Calculator

Easy #3

The answer is 223.

2 solutions

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