Question

How many positive integer less than 100 have the remainder of 2 when divided by 13?

Answer 1

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Answer 2

Answer:

The number of positive integers less than 100 and have remainder 2 when divided by 13 = 7

Step-by-step explanation:

To find,

The number of positive integers less than 100 and have remainder 2 when divided by 13

Solution:

The numbers which are multiples of 13 are of form 13n, where 'n' is a natural number

Hence the numbers which give remainder 2, when divided by 13 is of the form, 13n+2

Let 'S' be the set of numbers which leaves the remainder 2 when divided by 13

The first positive integer in the set S =  13n+2 = 13+2 = 15

The second positive integer in the set S= 13×2+2 = 28

The third positive integer in the set S = 41

Hence S = {15,28,41,........}

The greatest number less than 100 and is of the form 13n+2 is 93

Hence S = {15,28,41,.............93}

The elements of the set S form an AP with first term a = 15 and common difference =d = 13

Required to find,

The number of elements in the set S

we know, the number of elements of an AP

$n = \frac{t_n - t_1}{d} +1$

$t_n = 93, t_1 = 15,d =13$

$n = \frac{93 - 15}{13}+1$

= 6 +1

= 7

∴The number of positive integers less than 100 and have remainder 2 when divided by 13 = 7

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