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?
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?
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.
Hello, rcs!
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: .$\displaystyle 120 + 1 \:=\:121.$
Do you see why?