Question

16. Find the sum of all two-digit natural
numbers which when divided by 11 give
1 as remainder.​

Answer 1

Solution :

Two - digit natural numbers which are divisible by 11 are 11, 22, 33, 44, 55, 66, 77, 88, 99

So, Two - digit number when divided by 11 and give 1 as remainder are 12, 23, 34, 45, 56, 67, 78, 89

a₂ - a₁ = 23 - 12 = 11

a₃ - a₂ = 34 - 23 = 11

Since common difference is equal in every time the given numbers form an AP

There are 8 terms in AP

Number of terms ( n ) = 8

First term ( a ) = 12

Last term ( l ) = 89

Using Sum of n terms formula

S(n) = n / 2 ( a + l )

Sum of 8 terms = S₈ = 8 / 2 ( 12 + 89 )

⇒ S₈ = 4 × 101

⇒ S₈ = 404

Hence the sum of two - digit numbers when divided by 11 and leave 1 as remainder is 404.

Answer 2

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