# PROBABILITY on number of cards.

• Sep 7th 2012, 06:15 AM
rcs
PROBABILITY on number of cards.
Problem:
When john sorts his collection of computer card games into groups of 3, 4, 5 or 8, there is always one card left. what is the smallest number of cards John can have?
• Sep 7th 2012, 06:42 AM
HallsofIvy
Re: PROBABILITY on number of cards.
If we let N be the number of cards we are looking for then n= 3i+ 1, n= 4j+ 1, n= 5k+ 1, n= 8l+ 1. (This is equivalent to saying that N is "equal to 1 modulo 3, 4, 5, 8".)

We can, for example, set n= 3i+ 1= 4j+ 1= 5k+ 1= 8l+ 1 so that 3i= 4j= 5k= 8l. From "4j= 8l", we must have = 2l so we can reduce that to 3i= 5k= 8l. Now, what are the smallest possible values of i, k, and l so that those are true? That is, what is the smallest number that k is a multiple of 3, 5, and 8?
• Sep 7th 2012, 06:57 AM
rcs
Re: PROBABILITY on number of cards.
Im just so confused, according to the book from i got this problem is 121, i dont know how the author obtained it
• Sep 7th 2012, 07:10 AM
HallsofIvy
Re: PROBABILITY on number of cards.
You have posted a large number of widely varying problems without showing any attempt of your own on any of them. As far as this problem is concerned, if you cannot find "the smallest number that k is a multiple of 3, 5, and 8", you should not even be attempting these problems.

Do you see that 3, 5, and 8 have no divisors in common? So knowing that what is the smallest number that has is a multiple of 3, 5, and 8.
• Sep 7th 2012, 08:42 AM
Soroban
Re: PROBABILITY on number of cards.
Hello, rcs!

Quote:

When John sorts his collection of computer card games into groups of 3, 4, 5 or 8,
there is always one card left. .What is the smallest number of cards John can have?

The least common multiple (LCM) of {3, 4, 5, 8} is 120.

The number of computer card games is: . $120 + 1 \:=\:121.$

Do you see why?
• Sep 8th 2012, 05:11 AM
rcs
Re: PROBABILITY on number of cards.
thanks a lot Prof. Soroban. You are a big Help :)
• Sep 8th 2012, 05:48 AM
HallsofIvy
Re: PROBABILITY on number of cards.
Quote:

Originally Posted by rcs
thanks a lot Prof. Soroban. You are a big Help :)

Could you answer his question, then? Why is 121 the correct answer and what does this problem have to do with the "least common multiple" of 3, 5, and 8.