Kenneth Rosen Edition 7 Exercise 6.1 Question 21 (Page No. 396)

Question

How many positive integers between $50$ and $100$

• A. are divisible by $7?$ Which integers are these?
• B. are divisible by $11?$ Which integers are these?
• C. are divisible by both $7$ and $11?$ Which integers are these?

Answer 1

Solved by using very intuitive formula :

$ \left \lfloor \frac{b}{n} \right \rfloor-\left \lfloor \frac{a-1}{n} \right \rfloor$

a & b are inclusive by default 

Here,

(A) a=50, b=100, n=7

(B) a=50, b=100, n=11

(C) a=50, b=100, n=77(l.c.m of 7 & 11)

 

Please don't worry 

If you're not getting this formula as of now. I've attached the detailed solution for this question in which intuition behind this formula mentioned in a  pictorial way. I’m sure you'll definitely get this after going through the entire solution.

Please don't by heart this formula 

For writing this formula, you just to need to have  the idea of Complement Rule. That's it:)

 

Comments

That’s correct. 

Actually the formula for number of integers divisible by $n$ between $a$ and $b$ is $\displaystyle \left\lfloor \frac{b}{n} \right\rfloor – \left\lfloor \frac{a-1}{n} \right\rfloor $ if between is taken as “inclusive” which by default should be the case. But do watch out if “exclusive” is given in question. 

https://english.stackexchange.com/questions/7871/between-a-and-b-or-from-a-to-b

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@Manisha_K Suppose, I don’t know have an idea of complement rule and $\lfloor \frac{something}{n} \rfloor$

then is it possible for me to solve (A) and get a formula for (A) to get how many positive integers between $x$ and $y$ that are divisible by integer $m$ ?

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" Saying “between 1 and 10” is somewhat ambiguous; usually people will say “between 1 and 10 inclusive” or “between 1 and 10 exclusive” to clarify when there is no other context. Both “between…and…” and “from…to…” are usually considered inclusive unless otherwise specified. " ---> Now, the idea of exclusive is also clear in my mind.

Thank you sir:)

Answer 2

A. are divisible by 7 ? Which integers are these?

    Since it is between 50 and 100 only, i go by 7 multiple i.e.,

    7 x 8 =56

    7x 9 = 63

    7 x 10 = 70

    7 x 11 = 77

    7x 12  = 84

    7 x 13 = 91

    7 x 14 = 98.

   Therefore, the answer is 7 integers and they are 56, 63, 70, 77, 84, 91 and 98.

B.are divisible by 11? Which integers are these?

 Now go for 11 multiple i.e., 

11 x 5 = 55

11 x 6 = 66

11 x 7 = 77

11 x 8 = 88

11 x 9 = 99

 Therefore, the answer is 5 integers and they are 55, 66, 77, 88, 99.

C.are divisible by both 7 and 11? Which integers are these?

Now check the intersection of divisible by 7 and divisible by 11, we can find one integer which is common i.e., 77.

 Therefore, the answer is 1 and the integer is 77.

 

 

    

 

 

 

 

Comments

yes agreed.