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I am dealing with divisibility problems, such as:
Find the number of positive integers that is divisible by either of 2 or 5 within 1000. In this case, I decide to look at how many integers are divisible by 2 or 5 within 10, that is 6. So we can deduce that there are 600 numbers that could be divided by 2 or 5 within 1000.
But, what if we are trying to find number of positive integers that is divisible by both 3 or 7 within 1000, I found that the number block of '10' cannot work here, and if we try number block of '21', meaning that to find how many number is divisible by 3 or 7 within 21, and then use 1000 divides 21 to see how many blocks are there within 100, that will work. May I know what is the mechanism behind this or is there any general rule for dealing with this type problem?

Thank you so much for you guy's replies

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  • $\begingroup$ There are $\lfloor\frac{1000}2\rfloor+\lfloor\frac{1000}3\rfloor-\lfloor\frac{1000}6\rfloor=667$ numbers between $1$ and $1000$ divisible by $2$ or $3$. $\endgroup$ – saulspatz Feb 1 '20 at 16:44
  • $\begingroup$ Sorry, sorry, is not divisible by 2 or 5... $\endgroup$ – Henry Cai Feb 1 '20 at 16:48
  • $\begingroup$ @saulspatz hello sir, your method can also be used. Thank you very much. This is much help. $\endgroup$ – Henry Cai Feb 1 '20 at 16:49
  • $\begingroup$ Sure I will change it now $\endgroup$ – Henry Cai Feb 1 '20 at 16:54
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The size of the number block is the least common multiple of the divisors. If the divisors do not have a common factor it is just the product, but if your divisors are $4,6$ the minimum block would be $12$. It is the shortest block which goes through an integer number of cycles of all the divisors.

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  • $\begingroup$ Thank you very much sir $\endgroup$ – Henry Cai Feb 1 '20 at 16:53

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