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sriramsundaram91
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Bunuel
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freebunny

This may sound a very very stupid question, but I want to ask, how do you figure out multiples of 2 and 5 in a given range? Like you have mentioned:

Multiples of 2 in the range 0-1000, not inclusive - \(\frac{998-2}{2}+1=499\);

Is there any formula for this? Or is it just some logic that I am unable to figure out?


\(Number \ of \ multiples \ of \ x \ in \ the \ range = \)

\(=\frac{Last \ multiple \ of \ x \ in \ the \ range \ - \ First \ multiple \ of \ x \ in \ the \ range}{x}+1\).

Example 1: how many multiples of 5 are there between -7 and 35, not inclusive?

Last multiple of 5 IN the range is 30;
First multiple of 5 IN the range is -5;

\(\frac{30-(-5)}{5}+1=8\).

Example 2: How many multiples of 4 are there between 12 and 96, inclusive?
\(\frac{96-12}{4}+1=22\).

Example 3:
How many multiples of 7 are there between -28 and -1, not inclusive?
Last multiple of 7 IN the range is -7;
First multiple of 7 IN the range is -21;

\(\frac{-7-(-21)}{7}+1=3\).
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How about using Euler's totient formula? we can solve such problems quickly.

f(n)=n*(1-(1/p1))*(1-(1/p2))*.......* (1-(1/pn)) where p1 to pn are the prime factors of n.

1000= 2^3*5^3
f(1000)= 1000*(1-(1/2))*(1-(1/5))
=1000*1/2*4/5=400
Bunuel


First of all it should be "how many positive integers less than 1000 have no factors (other than 1) in common with 1000", as if we consider negative integers answers will be: infinitely many.

\(1000=2^3*5 ^3\) so basically we are asked to calculate the # of positive integrs less than 1000, which are not multiples of 2 or/and 5.

Multiples of 2 in the range 0-1000, not inclusive - \(\frac{998-2}{2}+1=499\);
Multiples of 5 in the range 0-1000, not inclusive - \(\frac{995-5}{5}+1=199\);
Multiples of both 2 and 5, so multiples of 10 - \(\frac{990-10}{10}+1=99\).

Total # of positive integers less than 1000 is 999, so # integers which are not factors of 2 or 5 equals to \(999-(499+199-99)=400\).

Answer: A.
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