Yan Cui

I help clients go faster for less using serverless technologies.

**This article is brought to you by**

Don’t reinvent the patterns. Catalyst gives you consistent APIs for messaging, data, and workflow with key microservice patterns like circuit-breakers and retries for free.

#### Problem

The fraction^{49}/_{98}is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that^{49}/_{98}=^{4}/_{8}, which is correct, is obtained by cancelling the 9s.

We shall consider fractions like,^{30}/_{50}=^{3}/_{5}, to be trivial examples.

There are exactly four non-trivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.

If the product of these four fractions is given in its lowest common terms, find the value of the denominator.

#### Solution

// define function to check if two numbers has the cancelling behaviour let isCancelling a b = let aStr, bStr = a.ToString(), b.ToString() if aStr.[0] = bStr.[0] then double(aStr.[1].ToString())/double(bStr.[1].ToString()) = double(a)/double(b) else if aStr.[0] = bStr.[1] then double(aStr.[1].ToString())/double(bStr.[0].ToString()) = double(a)/double(b) else if aStr.[1] = bStr.[0] then double(aStr.[0].ToString())/double(bStr.[1].ToString()) = double(a)/double(b) else if aStr.[1] = bStr.[1] then double(aStr.[0].ToString())/double(bStr.[0].ToString()) = double(a)/double(b) else false // define function to find the greatest common dividor let gcd a b = [2..min a b] |> List.rev |> Seq.filter (fun x -> a % x = 0 && b % x = 0) |> Seq.head // define function that returns numbers >= n which are not multiples of 10 // and have two distinct digits (i.e. 11, 22 only have one distinct digits) let numbers n = [n..99] |> List.filter (fun x -> x % 10 <> 0 && (x % 10 <> x/10)) let answer = // first work out the product of all 4 fractions, in fractions form (num, denom) let fraction = numbers 10 |> List.collect (fun x -> numbers (x+1) |> List.map (fun y -> (x, y))) |> List.filter (fun (x, y) -> isCancelling x y) |> List.reduce (fun (num, denom) (x, y) -> (num*x, denom*y)) // then define the denominator by the greatest common divisor (snd fraction) / gcd (fst fraction) (snd fraction)

**Whenever you’re ready, here are 3 ways I can help you:**

**Production-Ready Serverless**: Join 20+ AWS Heroes & Community Builders and 1000+ other students in levelling up your serverless game. This is your one-stop shop for**quickly levelling up your serverless skills**.- I help clients
**launch product ideas**,**improve their development processes**and**upskill their teams**. If you’d like to work together, then let’s**get in touch**. **Join my community on Discord**, ask questions, and join the discussion on all things AWS and Serverless.