Math Help - Divide by 9 Question

1. Divide by 9 Question

Hi All,

I am struggling with this question.

How many integers from 1 to 100 are divisible by 9? How many integers from 1 to 2000 are divisible by 9? If n, a are positive integers, how many integers from 1 to n are divisible by a?

Is there a rule that can help me with this??

Thanks

2. Originally Posted by Joel
If n, a are positive integers, how many integers from 1 to n are divisible by [I]a?
Hi Joel.

The answer is $\left\lfloor\frac na\right\rfloor$ where $\lfloor x\rfloor$ denotes the greatest integer not greater than $x.$

Proof:

Use the division algorithm and write $n=qa+r$ where $0\le r Now any integer divisible by $a$ is of the form $ka$ for some integer $k.$ We have

$1\le ka\le n$

$\implies\ ka\le qa+r

$\implies\ k

$\implies\ k\le q$

Conversely $k\le q\ \implies\ 1\le ka\le qa.$ Hence there are exactly $q$ integers from 1 to $n$ which are divisible by $a.$

And $\left\lfloor\frac na\right\rfloor=\left\lfloor q+\frac ra\right\rfloor=q$ as $q$ is an integer and $0\le\frac ra<1.$ This completes the proof.

3. Originally Posted by Joel
How many integers from 1 to 100 are divisible by 9? How many integers from 1 to 2000 are divisible by 9? If n, a are positive integers, how many integers from 1 to n are divisible by a?
How many multiples of 9 are 100 or less? Divide 100 by 9 to get 11.111.... Then 9(11) = 99 is the largest multiple of 9 less than 100, and there are 11 multiples of 9 less than 100.

How many multiples of 9 are 2000 or less? Divide 2000 by 9 to get 222.222... Then 9(222) = 1998 is the largest multiple of 9 less than 2000, and there are 222 multiples of 9 less than 2000.

How many multiples of 9 are n or less? This is where you have to use that funky function thing. Same as for "a" and n. But you can see the general principal above.

Note: If you're familiar with calculators, you might also refer to the "integer" function, which gives the whole-number part of the above divisions.

4. Thanks heaps. Your post not only gave me the answer, but when I am asked why this works in the future...thanks to you I have to tools to prove it. Cheers