Question: How many positive integers less than 1000 are multiples of 5 and are equal to 3 times an even number?
So Multiples of $5$ and $6$
If a number is a multiple of $5$ and $6$ then it is a multiple of $30$ as well. Because of the law,
Multiple of $a$ and $b$ $\implies$ multiples of lcm$(a, b)$
lcm($5, 6$) = $30$
$$1000 \equiv 10 \pmod{30}$$
I am trying to solve this problem using number-theory, any help? The answer is $33$
Also, how does $1000/30$ give the right answer?
Thanks!
Best Answer
As you have stated yourself, the integers we are looking for must be a multiple of $30$. Now it is simply a matter of counting all the multiples of $30$ below $1000$, which is found by dividing $1000$ by $30$ and take the integer part of the answer. That would be $33$.
To be more explicit, the numbers we are looking for are of the form $30k$ with $k \in \mathbb{N}$ such that $30 k <1000$. The numbers $k$ therefore have an upper limit that is governed by $k < 1000/30$. Since $k$ is an integer number, we might just as well rewrite this as $$ k \leq \mbox{floor}(1000/30)=33, $$ where floor means to take the integer part of a number. In this case floor$(1000/30) = $floor$(33.333) = 33$. Since $k$ should at least be $1$, we have $1 \leq k \leq 33$, which means there are $33$ numbers satisfying our requirements.