P3 (Challenge Problem) Avagadro\'s constant A, the number of molecules in 1 mole

Question

P3 (Challenge Problem) Avagadro's constant A, the number of molecules in 1 mole of a substance, is taken to be 6.022 times 1023. What is the last digit of the Ath Fibonacci number F6.022 times 1023 (written as a decimal number)? What are the last two digits? [Note: Challenge problems are optional. No credit is offered. You are requested to solve the remaining problems before attempting this.] [Hint: First observe the pattern of last digits of Fibonacci numbers and see if it repeats. If you do this by hand, then you have to be patient. Next, apply the fact that the pattern of last digits repeat in a cyclic fashion to figure out what the last digit must be.]

Explanation / Answer

By searching the internet you see that the sequence of final digits in Fibonacci numbers repeats in cycles of 60. The last two digits repeat in 300.

( Source : http://mathworld.wolfram.com/FibonacciNumber.html )

You can verify that with a computer program in any language you want for the first million of number.

Let's start with last digit :

6.022x10^23 = 6022x10^20

6022 = 22 mod 60

10^20 = 40 mod 60

So 6.022x10^23 = 22*40 mod 60 = 40

Fibonacci 40th number is 102334155 ( found by wolframalpha)

So the last digits seeked is 5

Now with the last two digits :

6022 = 22 mod 300

10^19 = 100 mod 300

So 6.022x10^23 = 100*22 mod 300 = 100 mod 300

Fibonacci 100th number is 354224848179261915075

So the last two digits are 75

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