Math Is Fun Forum

evene

Member

Registered: 2015-10-18

Posts: 272

Something not related to Geometry

How many different factors does 972 have?

How many different ways can 6 books be arranged on a shelf if one of the books, book X, has to be first?

The last time I tried a problem like this, I got a number way off. I'm also really bad at these kinds of problems!

Oh and please show your work... I want to see how you managed to solve these amazingly difficult problems

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Something not related to Geometry

How many different ways can 6 books be arranged on a shelf if one of the books, book X, has to be first?

Since the first book has to be X, then there are 5 x 4 x 3 x 2 x 1 = 120 ways to arrange the other 5.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

Re: Something not related to Geometry

hi evene,

How many different factors does 972 have?

There may be a formula for this but I don't know one.   But there's not so many that you cannot just count them all:

step 1 find the prime decomposition.  .... divide by 2 until it won't go any more .... then by 3 .....................

972 = 2 x 2 x 3 x 3 x 3 x 3 x 3

Step 2 then start a list with 1 also including the factor that pairs with 1:    1 x 972

Then 2 x 486

Then 3 x 324

Then 4 (think (2 x 2 ) x (3 x 3 x 3 x 3 x 3) so 4 x 243

not 5

Then 6 x 162

not 7, not 8

Then 9 x 108

not 10, not 11

Then 12 x 81

not 13, 14, 15, 16, 17

Then 18 x 54

not 19, ......26

then 27 x 36

not 28 ....35 so this is the complete list.

{1,2,3,4,6,9,12,18,27,36,54,81,108,162,243,324,486,972}

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

phanthanhtom

Member

Registered: 2012-06-22

Posts: 290

Re: Something not related to Geometry

Bob, you should be using combinatorics here. Since 972 = 2^2 × 3^5, each factor can only have prime factors 2 or 3.

Suppose factor n = 2^i × 5^j. We have 3 choices for i (0, 1, 2) and 6 choices for j (0, 1, 2, 3, 4, 5) and so 3×6=18 factors

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Something not related to Geometry

Thanks guys!