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Puzzle of the week: Clear the stage for arithmetic artists

2022-01-23T07:32:52.874Z


There are five numbers on the board. All are ten digits. But only one of them is the fourth power of a natural number. Which?


Enlarge image

Photo: DER SPIEGEL

In a quiz show, a candidate has to solve a complicated arithmetic problem.

You only have one minute to do this.

The showmaster briefly explains what to do:

“We'll show you a table with five ten-digit numbers in a moment.

Only one of them is the fourth power of a natural number.

You have one minute to find them!

Unfortunately, aids such as a pocket calculator are not permitted.«

"Hmm," says the woman, "how am I supposed to do that?" The quizmaster replies: "Unfortunately, I don't know either.

But you still have five minutes to think of something.

How long does the commercial break last?

When it's over, the plaque will be rolled onto the stage.«

The woman broods and broods - and then apparently has an idea.

When she is shown the five ten-digit numbers after the commercial break, she has found the one that is the fourth power of a natural number after just ten seconds.

How did she do that?

The solution is C).

The trick is to start by just looking at the ones digits of the five numbers.

The fourth power of an even natural number always ends in the number 6. You can see this immediately if you write the number to be raised in the form 10a + b.

The ones place of (10a + b)4 alone determines the ones place of b4.

In all other summands, 10 occurs as a factor.

Because the following applies: (10a + b)4 = 10000a4 + 4000a3b + 600a2b2 + 40ab3 + b4)

Because 24, 44, 64, and 84 all end in 6, so does the fourth power of any even natural number.

In the same way we find out that the fourth power of numbers ending in 1, 3, 7, 9 must have a 1 as the ones place.

If a number ends in 5, so does its fourth power.

If it ends in 0, there is also a 0 in the last place of the fourth power.

We summarize: The fourth power of a natural number ends in 0, 1, 5 or 6. This means that the numbers B), D) and E) are dropped.

Enlarge image

Photo: DER SPIEGEL

So there are only two numbers left as a solution: A) and C).

Now another special feature helps: The fourth power of an even number is always divisible by 16, because the following applies:

(2a)4 = 24a4 = 16a4

A number is divisible by 16 if its last four digits are a number divisible by 16.

Because multiples of 10,000 are always divisible by 16.

However, only the number C is divisible by 16).

Its last four digits are 8016. In A), only the last three digits are relevant, and 826 is not divisible by 16.

Therefore the solution can only be C).

In fact:

2524 = 4,032,758,016

Thanks to reader Thomas Grimm for suggesting this puzzle.

In case you missed a mystery from the past few weeks, here are the most recent episodes:

  • 18 dominoes for 54 squares

  • In the land of liars and truth fanatics

  • How big is the angle?

  • Fleeting encounter at the platform

  • The fly in the glass

  • How big is the yellow rectangle?

  • The main thing is 1000

  • heads or tails

  • freaky

  • How big is the yellow triangle?

Source: spiegel

All business articles on 2022-01-23

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