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Earlier as we speak I posted the next video, during which I requested Google Assistant to calculate the factorial of 100.
The factorial of 100 is the multiplication 100 x 99 x 98 x … x 3 x 2 x 1 during which 100 is multiplied by each entire quantity beneath it.
The reply is 158-digits lengthy. Google Assistant’s valiant effort, nevertheless, doesn’t get each digit right.
Immediately’s puzzle was:
What number of zeros does the factorial of 100 actually have on the finish of it?
Resolution:
[I will use the mathematical symbol ‘!’ for factorial below. Thus the factorial of 100 is also written 100!.]
I discussed within the authentic submit that if a quantity has a zero on the finish of it, it’s divisible by 10. What we have to do right here is figure out what number of occasions 10 divides into the quantity 100 x 99 x 98 x … x 3 x 2 x 1.
Let’s begin: 10 divides as soon as every into 10, 20, 30, 40, 50, 60, 70, 80, 90 and twice into 100, which suggests there should be no less than 11 zeros on the finish of 100!.
But it’s attainable to multiply two numbers that don’t finish in 0 to create one which does. For instance, 8 x 5 = 40. How can we account for all of the occasions that numbers within the breakdown of 100! multiply collectively to make a quantity divisible by 10?
The clue is to grasp that 10 = 2 x 5. And that each time two numbers multiply collectively to create a quantity that’s divisible by 10, there should be a 2 and a 5 concerned.
E.g. 8 x 5 = 2 x 2 x 2 x 5 = (2 x 2) x (2 x 5) = 4 x 10 = 40.
So, we are able to rephrase our activity as having to search for all of the situations of (2 x 5) in 100!. In different phrases, we have to break down each quantity from 1 to 100 into its components and see what number of occasions 5 and a pair of seem.
What number of occasions does 5 seem as an element within the numbers from 1 to 100? Nicely, counting upwards in 5s, we get 5, 10, 15, 20…90, 95, 100. These 20 numbers have 5 as an element. In actual fact 25, 50, 75 and 100 have 5 as an element twice. So the variety of occasions that 5 seems as an element is 24.
We are able to shortly see that 2 seems as an element no less than 24 occasions (simply depend the even numbers), so the whole variety of occasions (2 x 5) seems in 100! should be 24. The variety of zeros on the finish of 100! is 24.
For individuals who have an interest, 100! in all its glory is:
93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
Ta-dah!
I set a puzzle right here each two weeks on a Monday. I’m all the time on the look-out for excellent puzzles. If you want to recommend one, e mail me.
I’m the creator of a number of books of puzzles, most not too long ago the Language Lover’s Puzzle Book. I additionally give college talks about maths and puzzles (on-line and in particular person). In case your college is please get in contact.
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