Question

How many integers between 50 and 500 which are divisible by 7?

Answer 1

Hint: For solving this problem, we have to find out the first number after 50 and the last number before 500 which is divisible by 7. After finding these two numbers, we can apply the concept of arithmetic progression to calculate the number of terms. Using this methodology, we can easily solve the problem.

Complete step-by-step answer:
In mathematics, the number system is the branch that deals with various types of numbers possible to form and easy to operate with different operators such as addition, multiplication and so on. Integers are those numbers which are not in fraction and can occupy negative values.

First, dividing 50 by 7 leaves remainder 1. So, the first term after 50 which is divisible by 7 would be 56. Now, dividing 500 by 7 leaves a remainder 3. So, the last term would be 497.

Now, we have to find the total terms between 56 and 497 which are divisible by 7. To do so, we form an A.P. having a common difference of 7 with first and last term being 56 and 497 respectively.

So, by using the general term for an A.P.: ${{a}_{n}}=a+(n-1)\cdot d$

Now, putting values of variables as ${{a}_{n}}$ = 497, a = 56 and d = 7, we get

$\begin{align}

  & 497=56+(n-1)\cdot 7 \\

 & 497-56=(n-1)\cdot 7 \\

 & 441=(n-1)\cdot 7 \\

 & n-1=\dfrac{441}{7} \\

 & n-1=63 \\

 & n=64 \\

\end{align}$

Hence, there are 64 numbers between 50 and 500 which are divisible by 7.


Note: The key concept and solving this problem is the knowledge of arithmetic progression. For solving any problem in which a number of terms are required, we can form an A.P. and then apply the formula to evaluate the number of terms.