1
$\begingroup$

Is there a way to tell - for example - how many $60$s are there in a number that cannot be divided by $60$, like $183$? I know that we should remove all numbers after the decimal point but how can we tell this mathematically?

$\endgroup$
  • 1
    $\begingroup$ By inspection or some kind of algorithm? $\endgroup$ – Meow May 16 '13 at 17:13
  • $\begingroup$ Write $183=60\times k + r$ with $0< r < 60$. See here: en.wikipedia.org/wiki/Division_algorithm $\endgroup$ – Sigur May 16 '13 at 17:13
  • $\begingroup$ What do you mean "remove all numbers after the decimal point"? Your question is confusing. $\endgroup$ – 6005 May 16 '13 at 17:24
  • $\begingroup$ @Goos, he/she is saying that $183/60=3.05$ and if you take only the integer part you'll have the answer, that is, $3$. $\endgroup$ – Sigur May 16 '13 at 17:29
  • $\begingroup$ @Goos - I suspect the OP means that one divides the number by 60 on a calculator and takes the integer part of the quotient (at least that's how I interpreted it). I prefer the division algorithm approach as Sigur suggests, but I guess it all boils down to the same thing. $\endgroup$ – Chris Leary May 16 '13 at 17:32
3
$\begingroup$

It is still unclear whether you are looking for a trick, a formula, an algorithm, or a mathematical definition, so I will provide all four.

Trick: Probably the easiest way to do this by hand is do long division on $183 / 60$ but throw away the remainder (or remove all the numbers after the decimal point, if you prefer).

Formula: This is generally written $\left\lfloor \frac{183}{60} \right\rfloor$. Here $\lfloor x \rfloor$ is called the "floor" of $x$ and means the greatest integer less than or equal to $x$.

Algorithm: This is called the Division Algorithm (as was mentioned by Sigur in the comments). Basically, you subtract 60 from 183 until you get something on the interval [0, 60), and you count the number of times you subtracted 60 to get the answer.

Definition: Let $a$ and $b$ be integers. The integer quotient of $a$ and $b$ is defined to be the largest integer $k$ such that $a > bk$. (Note that this is well-defined if and only if $b > 0$). In the case of 183 and 60, we want the largest integer $k$ such that $183 > 60k$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.