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Today, let’s take a look at an interesting number property. Once we discuss it, you might think, “I always knew that!” and “Really, what’s new here?” So let me give you a question beforehand:

For integers and , . What is the value of ?

Think about it for a few seconds – could you come up with the answer in the blink of an eye? If yes, great! Close this window and wait for the next week’s post. If no, then read on. There is much to learn today and it is an eye-opener!

Let’s start by jotting down some powers of numbers:

Power of 2: 1, 2, 4, 8, 16, 32 …

Power of 3: 1, 3, 9, 27, 81, 243 …

Power of 4: 1, 4, 16, 64, 256, 1024 …

Power of 5: 1, 5, 25, 125, 625, 3125 …

and so on.

Obviously, for every power of 2, when you multiply the previous power by 2, you get the next power ().

For every power of 3, when you multiply the previous power by 3, you get the next power (), and so on.

Also, let’s recall that multiplication is basically repeated addition, so is basically 4 + 4.

This leads us to the following conclusion using the power of 2:

(2 times gives )

Similarly, for the power of 3:

(3 times gives )

And for the power of 4:

(4 times gives )

Finally, for the power of 5:

(5 times gives )

Quite natural and intuitive, isn’t it? Take a look at the previous question again now.

Question: For integers and , . What is the value of ?

A) 18

(B) 32

(C) 35

(D) 64

(E) 70

Which two powers when added will give ?

From our discussion above, we know they are and .

So and will satisfy this equation.

Therefore, our answer is **(E)**.

One question arises here: Is this the only possible sum of and ? Can and take some other integer values such that the sum of and will be ?

Well, we know that no matter which integer values and take, and will always be positive, which means both and must be less than 36. Now note that no matter which two powers of 2 you add, their sum will always be less than

. For example:

etc.

So if and are both integers, the only possible values that they can take are 35 and 35.

How about something like this: ? What integer values can , and take here?