So you need to make a 1:N gearing ratio for your project where N is some whole number like 21, 40, or even 100?  In general this can be easy or difficult depending on your number N. One general solution is to create a gear chain using n gearing ratios of 1:M0, 1:M1, 1:M2,... ,1:Mn. such that M0 x M1 x ... x Mn = N then the overall gear chain ratio will be 1:N. So that means you need to break down your number N into a number of factors that can be created by a pair of Lego gears. For example if N is 36 then you can write

36 = 3 x 3 x 2 x 2

So then you create a 1:36 gear ratio by creating a gear chain of simpler gears using 1:3, 1:3, 1:2 and 1:2. Now of course since the order of multiplication doesn't matter you can put these in any order, e.g. you could also use 1:2, 1:3, 1:3 and 1:2. So below is an example solution for a 1:36 gearing ratio using the first order shown.

Here the gearing ratio between the brown input axle and yellow output axle is 1:36.

Now due to the limited number of gears available in Lego Technic, it is only possible to create the gearings ratios of 1:2, 1:3, 1:5 and 1:7 directly using just two gears (i.e. without differentials) so that means your number N must be a product of one or more of the numbers 2, 3, 5 and 7 for this method shown here to work. For example below are the possible 1:N gearing ratios up to N <= 50 using this method:

Now of course you are not limited to just 1:M gearing ratios in the product, especially since 1:7 can only be made with an 8 tooth gear driving a 56 tooth turntable and so it is more convenient to use a 20 to 28 which gives 4:7 and you then need to cancel out the 4 by placing it in the denominator.

In summary, to create your 1:N gearing ratio, just look up your number in the table above and use the required factors (you will have to work them out for yourself for N > 50) in a gear chain.

Now what if you need to create a gearing ratio of 1:N where N cannot be created as a product of 2, 3 ,5 and 7 (e.g. 13, 17, 71 or other  primes)? The answer is to use a differential! (see the section of differentials for more details when that becomes available...).