What number do 21 28 and 35 all go into?

Answer 1

Do you mean: What is the smallest positive integer into which 21, 28, and 35 all divide evenly?

The answer is 420.

420 ÷ 21 = 20;

420 ÷ 28 = 15;

420 ÷ 35 = 12;

and all divide nicely, without remainder.

How did I find it? Did I try all the numbers, counting upwards until I found it? Of course, not. A mathematician is much too lazy for that. Always looking for the easy way out of honest, hard work, is the mathematician!

Now, just look here: Did you notice that

21 = 7 X 3;

28 = 7 X 4; and

35 = 7 X 5.

So, there is very likely some short cut. Now, that's using your head!

What's the smallest positive integer that 3, 4, and 5 divide into? It is 60 = 3 X 4 X 5. You are allowed to do that, since there is no number larger than 1 that divides into 3 and 4, or into 3 and 5, or into 4 and 5. (Thus, 3, 4, and 5 are what we call relatively prime - relatively to one another, that is. Any road, that's just a convenient name we use for such a set of numbers.)

Since 420 = 7 X 60, then 420 must be the least positive integer into which 7 X 3, 7 X 4, and 7 X 5, all three divide evenly.

By the by, all three of your original numbers, 21, 28, and 35, also divide evenly into 840 or 1260 or 1680 or 2100 … and so on; that is, into any multiple of 420. Can you see why?

Also, your three numbers divide equally well into 0, -420, -840, and so on.

But probably you wanted 420, the smallest such number that is positive. Then, it's easy to find the rest.

Now that was hardly any work, at all! The trick of maths is not piles and piles of hard work - no matter what your poor, hard-working teachers who hate maths, tell you. The trick is in first trying to understand the problem that you are solving. It is like solving any puzzle.