Question

How many three-digit numbers are divisible by 87 ?

Answer 1

This question can be solved by using the concept of arithmetic progression. The arithmetic progression has a sequence of the form $a,\left( a+d \right),\left( a+2d \right),\left( a+3d \right),\left( a+4d \right),\ldots \ldots$ Here, the first term of the sequence is given by a and the common difference of the consecutive terms is given by d. We calculate this series for all the 3-digit numbers which are divisible by 87 and count the number of terms by calculating n to get the answer.

Complete step by step solution:
The general form of an arithmetic progression is given by,
$\Rightarrow {{a}_{n}}=a+\left( n-1 \right)d$
Here, a represents the first term, d represents the common difference and n represent the nth term.
Using this we can solve the above question. It is required that all the 3-digit numbers are divisible by 87. The first 3-digit number divisible by 87 is 174. We know that the consecutive terms can be found by adding 87 to the number 174 first and then its result, etc. This is the same as arithmetic progression.
We need to substitute the values in the above equation and calculate the value of n to find out the 3-digit numbers which are divisible by 87. Here, we know that the highest 3-digit number divisible by 87 is 957. This can be used as the nth term which is ${{a}_{n}}$ in the equation. For this, we use the first term as the smallest 3-digit number divisible by 87 which is 174. Now the common difference is nothing but the value 87 itself. Plugging these values in the equation,
$\Rightarrow {{a}_{n}}=a+\left( n-1 \right)d$
Substituting ${{a}_{n}}$ as 957, a as 174 and d as 87,
$\Rightarrow 957=174+\left( n-1 \right)87$
Taking the product of the term in brackets with 87,
$\Rightarrow 957=174+87n-87$
Taking all the constant terms to one side,
$\Rightarrow 957-174+87=87n$
Adding and subtracting the terms on the left-hand side,
$\Rightarrow 870=87n$
Dividing both sides of the equation by 87,
$\Rightarrow \dfrac{870}{87}=n$
We simplify the division as,
$\Rightarrow 10=n$

Hence, the number of three-digit numbers are divisible by 87 are 10.

Note: We need to have a thorough understanding of the concept of arithmetic progression in order to solve this question easily. We can use the arithmetic progression as the series itself and calculate the value of each term and count the number of terms to get the solution. This is another method of solving.