How many times can 30! be divided by 10?

[YAMATO2201]

Does the following sentence sound perfectly natural to you?

How many times can 30! be divided by 10?

(my English translation of a problem from an entrance examination to a Japanese junior high school)

 

[emsr2d2]

The question is OK but I have no idea why there's an exclamation mark after "30".

 

[YAMATO2201]

The question is OK but I have no idea why there's an exclamation mark after "30".

30! = 1×2×3×4×5×···×28×29×30

https://en.wikipedia.org/wiki/Factorial

 

[SoothingDave]

You can divide something by 10 an infinite number of times.

 

[bubbha]

Yes, it sounds natural.

The assumption here, assuming it's a problem from number theory, discrete math, or combinatorics, is that only integers are involved.

 

[Rover_KE]

30! is factorial 30, which comes to 265,252,859,812,191,058,636,308,480,000,000.

Are Japanese junior high school students expected to know or be able to calculate that?:shock:

Perhaps the answer expected is

30!
10

 

[YAMATO2201]

Japanese junior high school students

Japanese elementary school students

(my English translation of a problem from an entrance examination to a Japanese junior high school)

 

[Rover_KE]

I can't help feeling you've mistranslated it.

I suggest you post the original Japanese in the Other Languages forum.

 

[bubbha]

30! is factorial 30, which comes to 265,252,859,812,191,058,636,308,480,000,000.

Are Japanese junior high school students expected to know or be able to calculate that?:shock:

Perhaps the answer expected is

30!
10

That figure ends with 7 zeros, so it can be divided by ten 7 times. So the answer sought is 7.

You don't have to multiply it all out. You just need to notice that the factors include 10, 20, and 30 (providing three of the zeros), as well as 5, 15 and 25 (providing four 5s), which when multiplied by another factor, 16, provide the remaining four zeros.

 

[YAMATO2201]

30! is factorial 30, which comes to 265,252,859,812,191,058,636,308,480,000,000.

The answer is 7.

1×2×3×4×···×29×30
=
(5×10×15×20×25×30)×(8×16)×(1×2×3×4×6×7×9×···)
=
[5×(2×5)×(3×5)×(4×5)×(5×5)×(5×6)]×[(2×2×2)×(2×2×2×2)] ×(1×2×3×4×6×7×9×···)
=
[5×(2×5)×(3×5)×(4×5)×(5×5)×(5×6)]×[(2×2×2)×(2×2×2×2)] ×(1×2×3×4×6×7×9×···)
=
(5×5×5×5×5×5×5)×(2×2×2×2×2×2×2)×(2×2×2×···×3×3×3×···×7×7×7×···)
=
[(2×5)×(2×5)×(2×5)×(2×5)×(2×5)×(2×5)×(2×5)]×(2×2×2×···×3×3×3×···×7×7×7×···)
=
(10×10×10×10×10×10×10)×(2×2×2×···×3×3×3×···×7×7×7×···)

The orange part can't be divided by 10 because it can't be divided by 5.

----------------------------------------------------------

How many times can 3000! be divided by 10?

The answer is 748.

 

[jutfrank]

The answer is 7.

To be precise, the correct answer to the question as it is written is given in post #4:

You can divide something by 10 an infinite number of times.

However, the real answer is: as many times as you can be bothered.

 

[YAMATO2201]

However, the real answer is: as many times as you can be bothered.

:up::up::up:

 

[SoothingDave]

The missing word is "evenly." How many times can it be divided evenly.

 

[Rover_KE]

Japanese elementary school students

You don't have to multiply it all out. You just need to notice that the factors include 10, 20, and 30 (providing three of the zeros), as well as 5, 15 and 25 (providing four 5s), which when multiplied by another factor, 16, provide the remaining four zeros.

The answer is 7.

1×2×3×3×···×29×30
=
(5×10×15×20×25×30)×(8×16)×(1×2×3×4×6×7×9×···)
=
[5×(2×5)×(3×5)×(4×5)×(5×5)×(5×6)]×[(2×2×2)×(2×2×2×2)] ×(1×2×3×4×6×7×9×···)
=
[5×(2×5)×(3×5)×(4×5)×(5×5)×(5×6)]×[(2×2×2)×(2×2×2×2)] ×(1×2×3×4×6×7×9×···)
=
(5×5×5×5×5×5×5)×(2×2×2×2×2×2×2)×(2×2×2×···×3×3×3×···×7×7×7×···)
=
[(2×5)×(2×5)×(2×5)×(2×5)×(2×5)×(2×5)×(2×5)]×(2×2×2×···×3×3×3×···×7×7×7×···)
=
(10×10×10×10×10×10×10)×(2×2×2×···×3×3×3×···×7×7×7×···)

The orange part can't be divided by 10 because it can't be divided by 5.

----------------------------------------------------------

How many times can 3000! be divided by 10?

The answer is 748.

Japanese elementary students are expect to figure all that out?:shock:

Give me a break!

 

[YAMATO2201]

Japanese elementary students are expect to figure all that out?:shock:

Only those who are going to take an entrance examination to a prestigious private Japanese junior high school.

(Do you mean "... are expected to ..."?)

-----------------------

[STRIKE]1×2×3×3×···×29×30 [/STRIKE] 1×2×3×4×···×29×30

Sorry for the typo.

 

[Charlie Bernstein]

You can divide something by 10 an infinite number of times.

You beat me to it, Dave. You can divide thirty by ten all day long.

Better phrasing is:

- How many times does ten go into thirty?
- What is thirty divided by ten?

 

[Rover_KE]

But those simple questions aren't the ones those genius Japanese elementary children have to answer!

 

[Tdol]

How many times can 3000! be divided by 10?

The answer is 748.

Your children have it hard in maths classes.

 

[Charlie Bernstein]

The answer is 7.

1×2×3×4×···×29×30
=
(5×10×15×20×25×30)×(8×16)×(1×2×3×4×6×7×9×···)
=
[5×(2×5)×(3×5)×(4×5)×(5×5)×(5×6)]×[(2×2×2)×(2×2×2×2)] ×(1×2×3×4×6×7×9×···)
=
[5×(2×5)×(3×5)×(4×5)×(5×5)×(5×6)]×[(2×2×2)×(2×2×2×2)] ×(1×2×3×4×6×7×9×···)
=
(5×5×5×5×5×5×5)×(2×2×2×2×2×2×2)×(2×2×2×···×3×3×3×···×7×7×7×···)
=
[(2×5)×(2×5)×(2×5)×(2×5)×(2×5)×(2×5)×(2×5)]×(2×2×2×···×3×3×3×···×7×7×7×···)
=
(10×10×10×10×10×10×10)×(2×2×2×···×3×3×3×···×7×7×7×···)

The orange part can't be divided by 10 because it can't be divided by 5.

----------------------------------------------------------

How many times can 3000! be divided by 10?

The answer is 748.

I have absolutely no idea what any of that means.

The world is lucky I don't give math advice!

 

[jutfrank]

Yamato2201,

Have you Japanese people never heard of the calculator?