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 Unearthing Units Digit (Posted on 2016-08-09)
Find the units digit of floor(N), where:
```    1020000
N = ---------
10100 + 3```
*** For an extra challenge, solve this puzzle without using a computer program aided method.

 No Solution Yet Submitted by K Sengupta No Rating

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 Solution Comment 1 of 1
I'm going to generalize this a little bit and make
N = 10^(2*h^2)  / (10^h + 3)
For this problem h=100

Let M = 3^(2h)/(10^h + 3).  M is less than 1.  I will show that N-M is an integer, which means its last digit is the value KS asks us to find.

N-M = (10^(2*h^2) - 3^(2h))  / (10^h + 3)

Rearrange the exponents a little:
N-M = ((10^(2*h))^h - (3^2)^h))  / (10^h + 3)

This makes it easy to see that the numerator is of the form x^n-y^n.  Then the numerator can be factored.
N-M = [10^(2*h) - (3^2)] * [(10^(2*h))^(h-1) + (10^(2*h))^(h-2)*(3^2) + ... + (10^(2*h))*(3^2)^(h-2) + (3^2)^(h-1)]  / (10^h + 3)

The first term 10^(2*h) - (3^2) can be factored into (10^h - 3)*(10^h + 3).  The second factor cancels the denominator in the fraction, leaving
N-M =  (10^h - 3) * [(10^(2*h))^(h-1) + (10^(2*h))^(h-2)*(3^2) + ... + (10^(2*h))*(3^2)^(h-2) + (3^2)^(h-1)]

Since there are no more fractions, N-M must be an integer for positive integer h.  The units digit is equal to (-3) * (-1)^(h-1) mod 10.  This means when h is even the units digit is 3 and when h is odd the units digit is 7.  h=100 is even, so the digit KS asks for is 3.

 Posted by Brian Smith on 2017-05-05 23:23:10

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