What is the Units Digit?

#BinomialThoerem #Algebra #MathOlympiad

What is the units digit of this number?

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I share Maths problems and Maths topics from well-known contests, exams and also from viewers around the world. Apart from sharing solutions to these problems, I also share my intuitions and first thoughts when I tried to solve these problems.

[pbj]

This is a clever use of elementary properties. Liked and shared!

[ShinichiKudou]

Let a=45+√2020, and b=45-√2020.
We have ab=5 and a+b=90.
So we have a Lucas sequence with s(0)=2, s(1)=90 and s(n)=90s(n-1)-5s(n-2).
Obviously, all subsequent terms end with 0.
With a²⁰²¹+b²⁰²¹=0(mod 10) and 0<b<1, it is clear that a²⁰²¹ must end with 9 point something.
So the units digit is 9.

[DaveyJonesLocka]

Excellent job. I love your videos. I’m surprised no one pointed out the error in your third term in the binomial expansion of (45+SQRT(2020))^2021, where you wrote 45C2 instead of 2021C2. But, the property of that third term (it ends in zero) was not impacted.

Not being nit picky or critical. It is comforting knowing that the best mathematicians do make casual mistakes. Please keep up the excellent work.

[raffaelevalente]

I found the result with the technique of the "hazardous induction" :) Tried a few low exponents, get always 9. Then must be 9. DON'T DO THIS IN AN EXAMINATION!!!

[tieliu]

Very well-designed problem. The use of Galois conjugates in your solution is nice too. This is in spirit very similar to (but a bit easier than) one of the 1988 IMO shortlist problems.

[yoav]

nice trick, simple to understand but yet beautiful

[ncantor]

More generally:
Let's show by strong induction that for all non-negative integers a, b, n,
(a + √b)^n + (a - √b)^n = N_n which is a natural number.

For n = 0: N_0 = 2
For n = 1: N_1 = (a + √b) + (a - √b) = 2a which is clearly a N.N.
By the induction hypothesis:
(a + √b)^n + (a - √b)^n = N_n
2a[(a + √b)^n + (a - √b)^n] = 2aN_n
[((a + √b) + (a - √b)][(a + √b)^n + (a - √b)^n] = 2aN_n
(a + √b)^(n+1) + (a - √b)^(n+1) + (a - √b)(a + √b)^n + (a + √b)(a - √b)^n = 2aN_n
N_(n+1) + (a - √b)(a + √b)(a + √b)^(n-1) + (a + √b)(a - √b)(a - √b)^(n-1) = 2aN_n
N_(n+1) + (a - √b)(a + √b)[(a + √b)^(n-1) + (a - √b)^(n-1)] = 2aN_n
N_(n+1) + (a - √b)(a + √b)[N_(n-1)] = 2aN_n
N_(n+1) + (a^2 - b)N_(n-1) = 2aN_n
N_(n+1) = 2aN_n - (a^2 - b)N_(n-1)
The RHS is clearly a natural number by the strong induction hypothesis, so the LHS is as well.
Let's plug in numbers and calculate the ones digit for N_(n+1) by taking it mod 10. Note that N_0 = 2 and N_1 = 90
N_(n+1) = 2*45*N_n - (45^2 - 2020)N_(n-1) = 0 - (5^2 - 0)N_(n-1) = 5N_(n-1) [mod 10]
Note that the ones digit of N_2 then is 0 as is N_1. And by this recurrence, for all n >=1 the ones digit is going to be zero as well.
Then, note that (a - √b) < 1 and so (a - √b)^n is < 1, so the ones digit of (a + √b)^n is one less than that for N_n, so always 9 for n >=1.

This helps to find the ones digit for other values as well, even when b does not end in 0
So if b = 2021 instead, we see that:
N_(n+1) = 2*45*N_n - (45^2 - 2021)N_(n-1) = 0 + (1 - 5^2)N_(n-1) = 6N_(n-1) [mod 10]
We can see that the ones digits of N will repeat as n increases like:
2, 0, 2, 0, 2, 0, 2, 0, ...
Since (a - √b) < 1, the ones digit of (a + √b)^n repeats with 1 for even terms and 9 for odd terms.

[sasharichter]

Let x = 45 - Sqrt(2020) and y = 45 + Sqrt(2020) so that x + y = 90 and x*y = 5. Let A[n] = x^n + y^n then A[0] = 2, A[1] = 90  
and A[n] satisfies the standard recursion A[n] = (x + y)*A[n-1] - x*y*A[n-2] giving A[2] = 8090. Thus A[n] = 0 mod(10), n > 0.
Finally y^2021 = A[2021] - x^2021 = 0 - x^2021 mod(10) = 9 mod(10) since x < 1. Thus the unit digit is 9!

[HagenvonEitzen]

Well, (45+\sqrt(2020))^2021 + (45-\sqrt(2020))^2021 is an integer and the second number is very small and all non.cancelling summands are multiples of 5 and occur in both parts, so ...

[danielsittner]

Make a video on the latest Putman exam

[nezamasarie]

I just thought it was 5
Cause 2020^n will have a units digit of 0
And if u add 45 u have a units number of 5

[gamedepths]

If the number under the square root did not end with a 0, could we still find the units digit?

[tonyhaddad]

Why you dont say yay anymore ??? 😥😥😥😥😥😥😥