Numbers which are divisible

Toast · 2007-01-09 20:08:04

Question
How many numbers between 100 and 500 (inclusive) are divisible by either 4 or 5?

The correct answer is 161 numbers. I am out by just one, with an answer of 160. Can someone please correct my working so it gives the correct answer. Thanks.

Solution:
Let n(A) = numbers divisible by 4
Since (500÷4)-(99÷4) = 100.25, there are 100 numbers between 100 and 500 divisible by 4.
i.e. n(A) = 100.

Let n(B) = numbers divisible by 5
Since (500÷5)-(99÷4) = 80.2, there are 80 numbers between 100 and 500 divisible by 5.
i.e. n(B) = 80.

To find n(A and B), notice that if a number is divisible by both 4 and 5, it must also be divisible by 20.
Since (500÷20)-(99÷4) = 20.05, there are 20 numbers between 100 and 500 divisible by 20.

Using n(A or B) = n(A) + n(B) - n(A and B) gives
n(A or B) = 100 + 80 - 20
∴ n(A or B) = 160
Hence, there are 160 numbers between 100 and 500 divisible by either 4 or 5.

MathsIsFun · 2007-01-09 20:55:54

Not a full analysis of your approach, just your first part:

How many numbers are divisible by 4 between 16 and 20?

Since (20÷4)-(15÷4) = 1.25, hence 1 or 2?

Toast · 2007-01-09 21:26:40

Oh noes it doesn't make any sense!

mathsyperson · 2007-01-09 23:13:07

It makes perfect sense. Because of MathsIsFun's discovery, you need to add 1 to each of the numbers that you worked out.

So now your answer is 101+81-21 = 161.

Toast · 2007-01-10 12:43:30

Oh i see, so if I'm looking for integers between x and y, where x > 0, then I simply have to add one.

But what if I had a question, to say, find between 14 and 18 (inclusive) all integers divisible by 4:
(18 ÷ 4) - (13 ÷ 4) = 1.25. If I were to round this up as has been suggested it would be incorrect.

Toast · 2007-01-10 16:18:11

Oh I see, 99 ÷ 4 = 24.75. Rounded down = 24. If I had seperately worked that out instead of using the combined sum that I used, I would have gotten 101 for my answer.

Thanks everyone.

pi man · 2007-01-10 16:20:50

You would always round down. And it really makes mores sense if you round down the intermediate steps. Consider the 14 through 18 (inclusive) example. You divide 18 by 4 to find out how many multiples of 4 there are. You get 4 1/2 but you can't really have 1/2 of a multiple of 4. You really should round down to 4 here.

And dividing 13 by 4 gives you 3 1/3. Round it down to 3 and subtract it from 4 to get 1.

Edited: Looks like I'm a minute late on my reply!

Last edited by pi man (2007-01-10 16:22:15)

Toast · 2007-01-10 16:23:51

Thanks anyway pi man, I guess I got too used to doing it in one step, as I have learned to do with surds and trig to 'keep things accurate'

Toast · 2007-01-10 20:21:34

This is, in a way related,

Does 'int' in int(x) stand for integer? If it does then is x rounded up or down?

mathsyperson · 2007-01-10 20:52:40

Google says that the int stands for integral, and that int (f,x) is equivalent to saying ∫f dx.

Toast · 2007-01-10 20:54:15

Oh...in my book it says

int(1000 ÷ 7) = 142

MathsIsFun · 2007-01-10 22:25:39

Depends on the computer language, but usually int(x) returns the integer part, so int(12.2) = 12

Toast · 2007-01-10 22:44:20

If the x value's tenth's column is greater than or equal to 5, then is x rounded up, or is it always rounded down?

pi man · 2007-01-11 04:02:17

In the computer languages I've dealt with, the INT function always rounds down to the nearest integer. In other words, it truncates. If you wanted to round up any number with 5 or greater in the tenth's, you would do something like: INT (X+.5). If X was 12.1, X+.5=12.6, and the INT function would still chop it off to 12. If X was 12.6, X+.5 = 13.1 and the INT function would chop it off to 13, effectively rounding up.

Carly_baby_2 · 2007-01-13 05:49:50

Thats Correct! I have had trouble with that one myself!

pi man · 2007-01-13 07:36:56

And if you wanted you program to round to the nearest hundred, it would look something like this:
x = INT(x/100 + .5) * 100

Test Examples:
If x = 1949
x/100 = 19.49
x/100 + .5 = 19.99
INT (x/100 + .5) = 19
INT(X/100 + .5) * 100 = 1900

If x = 1950
x/100 = 19.5
x/100 + .5 = 20.0
INT(x/100 + .5) = 20
INT(x/100 + .5) * 100 = 2000

If x = 1951
x/100 = 19.51
x/100 + .5 = 20.01
INT(x/100 + .5) = 20
INT(x/100 + .5) * 100 = 2000

Toast · 2007-01-13 15:49:32

Oooh interesting, can't wait to learn computer programming.

Math Is Fun Forum

#1 2007-01-09 20:08:04

Numbers which are divisible

#2 2007-01-09 20:55:54

Re: Numbers which are divisible

#3 2007-01-09 21:26:40

Re: Numbers which are divisible

#4 2007-01-09 23:13:07

Re: Numbers which are divisible

#5 2007-01-10 12:43:30

Re: Numbers which are divisible

#6 2007-01-10 16:18:11

Re: Numbers which are divisible

#7 2007-01-10 16:20:50

Re: Numbers which are divisible

#8 2007-01-10 16:23:51

Re: Numbers which are divisible

#9 2007-01-10 20:21:34

Re: Numbers which are divisible

#10 2007-01-10 20:52:40

Re: Numbers which are divisible

#11 2007-01-10 20:54:15

Re: Numbers which are divisible

#12 2007-01-10 22:25:39

Re: Numbers which are divisible

#13 2007-01-10 22:44:20

Re: Numbers which are divisible

#14 2007-01-11 04:02:17

Re: Numbers which are divisible

#15 2007-01-13 05:49:50

Re: Numbers which are divisible

#16 2007-01-13 07:36:56

Re: Numbers which are divisible

#17 2007-01-13 15:49:32

Re: Numbers which are divisible

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