[Math] How I got this answer


Well i am new to this forum but old to stack exchange but i don't know much about mathmatics tags so sorry for the wrong tags if any.
I have a question, but i want to know how i got its answer as 'D'

Aslam has $4$ times as many books as Salman and $5$ times as many as Javed. If Javed has more than $40$ books, what is the least number of books that Aslam could have?

(A) $200$

(B) $205$

(C) $210$

(D) $220$

(E) $24$

Answer: $(D)$

Well I don't know how the answer is '$220$'. I am new to this field as well but according to me its answer should be $200$. Any ideas?


Best Answer

Since Javes has more than $40$ books, he has at least $41$ books. Thus, Aslam has at least $5\cdot 41=205$ books. Since his number of books is a multiple of $4$ and $5$, it is a multiple of $20$. Thus, he has at least $220$ books.

Say that $a$ is the number of books Aslam has. We know that $4|a$ ($4$ divides $a$) and that $5|a$. In general, if $k|x$ and $l|x$, it follows that $\text{lcm}(k,l)|x$, where $\text{lcm}(k,l)$ is the least common multiple of $k$ and $l$. Since the smallest number that is a multiple of both $4$ and $5$ is $20$, it follows that $20|a$.

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