Question

How many two digit numbers are divisible by 7?

Answer 1

Hint: To solve this problem, we will first start by enlisting the set of two digit numbers divisible by 7. We can then see that these numbers form an arithmetic progression with common difference. Thus, we will use the formula of arithmetic progression to solve the problem. The formula of the ${{n}^{th}}$ term of this series is given by -
${{a}_{n}}$= a + (n-1) d
${{a}_{n}}$= ${{n}^{th}}$ term of this series (in this case, highest two-digit number divisible by 7)
The value of n will give the required answer.

Complete step-by-step answer:
Since, we need to find the arithmetic progression to solve the problem, we first find the first term of the arithmetic progression. Thus, we first start by finding the smallest integer greater than 10 which is a multiple of 7. This is clearly 14 (since, 11,12 and 13 are not multiple of 7 and thus 14 would be the required number). Now, we find the last term. For this we start with 99 (largest number just smaller than 100). Since, this is not a multiple of 7, we then move onto 98. This is in fact a multiple of 7, thus we have found our last term. Since, we have to find the multiple of 7, the common difference is 7. Thus, we have in the equation ${{a}_{n}}$= a + (n-1) d, we have,
${{a}_{n}}$ = 98
a = 14
d = 7
Now, substituting the values, we have,
98 = 14 + 7 (n-1)
(n-1) = $\dfrac{98-14}{7}$
(n-1) = 12
n = 13
Thus, the number of two-digit numbers that are divisible by 7 are 13.

Note: Another way to find the answer between two numbers (say a and b) is by doing the following. We first find the remainder when we divide 100 by 7 (which is 2). We then subtract 2 from 100 to get the last term (100 – 2 = 98) of the arithmetic progression. We then find the remainder when we divide 10 by 7 (which is 3). We then add (7-3 = 4) to 10 (which is 10 + 4 = 14) to get the first term. Now, we find the answer by using the formula –
$\dfrac{98}{7}-\dfrac{14}{7}+1$ = 14 – 2 + 1 = 13 (which is the same answer as the one in the solutions.)