Question

How many 4 digit positive integers have the ones digit a multiple of 1, the tens digit a multiple of 2, the hundreds digit a multiple of 3 and the thousands digit a multiple of 4?

Answer 1

The multiples of four that occur between the digits 0-9 are 0, 4, and 8 -- but 0 would make it a 3 digit number, not a 4 digit integer so we throw that one out

The multiples of three that occur are 0, 3, 6, and 9
The multiples of two that occur are 0, 2, 4, 6, and 8
The multiples of one that occur are every number 0-9

So...
for the thousands we have 4 and 8
for the hundreds we have 0, 3, 6, and 9
for the tens we have 0, 2, 4, 6, 8
and
for the ones we have everything 0-9

now to determine how many numbers these conditions satisfy we multiply the amount of satisfactory amount of conditions for the thousands by the satisfactory conditions for the hundreds by the satisfactory conditions for the tens by the satisfactory conditions for the ones

There are two numbers that the thousands place can be -- 4 or 8
There are four numbers that the hundreds place can be -- 0, 3, 6 or 9
There are five numbers that the tens place can be -- 0, 2, 4, 6, or 8
and there are ten numbers that the ones place can be -- 0-9

2*4*5*10 = 4*100 = 400 different numbers