0

So the question is "How many numbers between $3$ and $101$ are exactly divisible by $4$?" I found out that the answer is $25$.

When reading this question over, a thought came into my head. What if the question said "are exactly multiples of $4$?"

Can someone explain the difference between the two in context of this problem?

Julian Rachman
  • 1,835
John
  • 159
  • The two statements are exactly the same. (See what I did there?) – Edward Jiang Jan 02 '15 at 20:56
  • I'm sorry, let me give a better example. So, would 100 be divisible by 10? Or is 10 a multiple of 100? – John Jan 02 '15 at 20:57
  • An integer number $a$ is divisible by the integer $b$ if and only if $a$ is a multiple of $b$. It's the same. So $100$ is a multiple of $10$ and $100$ is divisible by $10$. You're probably confused by “being a divisor”: $10$ is a divisor of $100$. – egreg Jan 02 '15 at 20:58
  • Is this correct?: 1000 is a multiple of 100. 100 is divisible by 10? – John Jan 02 '15 at 20:59
  • @John Why do you think they differ? Perhaps the problem may stem from thinking that $0$ or $4$ is not an "exact multiple" of $4$? What do you mean by an exact multiple and exactly divisible? – Bill Dubuque Jan 02 '15 at 21:03
  • Okay maybe this what you meant: 4,8,12,16,20,24,.... we see from this sequence that maybe a little weird if we say 4 is a multiple of 4 (meanwhile "8 is multiple of 4" is perfectly fine) .and also the word"multiple" itself means "MORE THAN ONE", i think thats exactly what makes mr John get confused – Gary B Jan 02 '15 at 21:12

1 Answers1

2

The two wordings are synonymous. "$a$ is a multiple of $b$" means neither more nor less than "$a$ is divisible by $b$".

... in the integers, that is. In general rings, "$a$ is divisible by $b$" will almost always mean that there is some $k$ in the ring such that $a=kb$, whereas "$a$ is a multiple of $b$" can sometimes (but not always; you'd need to research each author's usage to be sure) mean that $a$ is an integer multiple of $b$ -- that is, there exists an integer $n$ such that $a=nb$.

hmakholm left over Monica
  • 286,031