A number theory problem by Piyush Kumar

Number Theory Level 3

When 30! is computed it ends in 7 zeroes. find the digit that immediately precedes these zeroes.

The answer is 8.

2 solutions

Adarsh Kumar
Sep 14, 2014

$First,prime\ factorise\ 30!,you\ will\ get:$ $2^{26}*3^{14}*5^{7}*7^{4}*11^{2}*13^{2}*17*19*23*29$ $Which\ can\ be\ written\ as:$ $2^{19}*3^{14}*7^{4}*11^{2}*13^{2}*17*19*23*29*10000000$ $When\ you\ calculate\ the\ unit\ digit\ of:$ $2^{19}*3^{14}*7^{4}*11^{2}*13^{2}*17*19*23*29$ $You\ will\ get\ 8.$ $\Longrightarrow\ it\ is\ 80000000.$ $Thus,the\ first\ nonzero\ digit\ is\ 8$

Would you do the same, were it 2014 instead of 30??

Satvik Golechha - 6 years, 8 months ago

-_- Do you know of an easier waY? If yes, post it

Krishna Ar - 6 years, 8 months ago

I don't, I was asking Adarsh, if they knew.

Satvik Golechha - 6 years, 8 months ago

@Satvik Golechha – no I wouldn't do it this way if it were 2014 and I don't know any other way.Sorry.

Adarsh Kumar - 6 years, 8 months ago

what i came up with was like this

1x3x4x6x7x8x9 and i got 8 as the last digit (excluding 2 and 5 because they contribute for 0)

when i did 11x13x14x16x17x18x19 then again i got 8 as last digit ;similarly for the twenties. so i concluded 8 as answer.

Afreen Sheikh - 6 years, 4 months ago

@Afreen Sheikh – 8 * 8 * 8 ends with 2, isn't it so?

Vijay Bhaskar - 6 years ago

@Vijay Bhaskar – i agree with u...but i did consider the 22,25,12,15 part as well after that

Afreen Sheikh - 6 years ago

Actually I got a formula to do it ,

$\lfloor \frac{N}{5} \rfloor !×2^{\lfloor \frac{N}{5} \rfloor}×(N\ mod\ 5)!$

Where : $N \rightarrow$ The number $(N!)$ whose last non-zero digit we have to calculate.

Akshat Sharda - 5 years, 9 months ago

Vincent Miller Moral
Apr 21, 2015

prod = 1
for i in range(1,31):
    prod *= i
    while prod%10==0:
        prod /= 10
prod %= 10
print prod

A number theory problem by Piyush Kumar

The answer is 8.

2 solutions

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