**Find the Sum of all Multiples Between Two Numbers :**

The example problems given in this page will be much useful for the students who would like to know the method of finding the sum of all multiples between two numbers.

**Question 1 :**

How many three digit numbers are divisible by 7?

**Solution :**

The first three digit number is 100 and the last three digit number is 999

Now we should find the number of three digit terms which are divisible by 7.

105 is the first three digit number divisible by 7 and 994 is the last three digit number which is divisible by 7.

105 + 112 + ........... + 994

a_{n} = a + (n - 1) d

a = 105, d = 112 - 105 = 7

994 = 105 + (n - 1)7

994 - 105 = (n - 1)7

889 = (n- 1)7

889/7 = n - 1

127 = n - 1

n = 127 + 1

n = 128

Hence there are 128 three digit numbers are divisible by 7.

**Question 2 :**

How many multiples of 4 lie between 10 and 250?

**Solution :**

12, 16, 20,.......................,248

a_{n} = a + (n - 1) d

a = 12 d = 16 - 12 = 4 a_{n} = 248

248 = 12 + (n - 1) 4

248 - 12 = (n - 1) 4

236 = (n - 1) 4

236/4 = n - 1

59 = n - 1

n = 59 + 1

n = 60

**Question 3 :**

For what value of n, are the nth terms of two APs 63, 65, 67,....... and 3, 10, 17,........ equal?

**Solution :**

Let a_{n} and b n are two n^{th} terms of the given sequence respectively.

a a = 63, d = 65 - 63 = 2 a |
b b = 3, d = 10 - 3 = 7 b |

(1) = (2)

a_{n} = b_{n}

63 + (n - 1) 2 = 3 + (n - 1) 7

63 + 2 n - 2 = 3 + 7n - 7

61 + 4 = 7 n - 2 n

65 = 5 n

n = 65/5

n = 13

After having gone through the stuff given above, we hope that the students would have understood, find the sum of all multiples between two numbers.

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