Question

#Quality Question
@ Sequence and Series

The Sum of all 2 digit positive numbers which when divided by 7 yeild 2 or 5 as remainder is ...​

Answer 1

ANSWER:

To Find:

• Sum of all 2-digit positive numbers which when divided by 7 yield 2 or 5 as remainder.

Solution:

We need to find the sum of all 2-digit positive numbers which when divided by 7 yield 2 or 5 as remainder.

Let the multiples of 7 be represented as 7k, where k ≥ 1.

Now, we will first find the sum of all 2-digit positive numbers which when divided by 7 yield 2 as remainder.

So,

⇒ The numbers = 7k + 2

As, 7k will be multiples of 7 and 2 will be the remainder.

Now, we'll find the numbers.

Putting k=1, we get 9. But, as 9 is a single-digit no. so we'll ignore this.

Putting other values of k(2 - 13), we get,

⇒ 16, 23, 30,. . . . , 93

We can see that this forms an AP, with first term 16, common difference 7 and last term 93.

We know that,

⇒ S = n/2 × (2a + (n - 1)d)

Or,

⇒ S = n/2 × (a + l)

Where, n is no. of terms, a is first term and l is last term.

So,

⇒ S = 12/2 × (16 + 93)

⇒ S = 6 × 109

⇒ S = 654 ----(1)

Now, taking remainder as 5.

So,

⇒ The numbers = 7k + 5

As, 7k will be multiples of 7 and 5 will be the remainder.

Now, we'll find the numbers.

Putting k=1, we get 12.

Putting other values of k(2 - 13), we get,

⇒ 19, 26, 33,. . . . , 96

Hence,

⇒ 12, 19, 26, 33,. . . . , 96

We can see that this forms an AP, with first term 12, common difference 7 and last term 96.

We know that,

⇒ S' = n/2 × (2a + (n - 1)d)

Or,

⇒ S' = n/2 × (a + l)

Where, n is no. of terms, a is first term and l is last term.

So,

⇒ S' = 13/2 × (12 + 96)

⇒ S' = 13/2 × 108

⇒ S' = 13 × 54

⇒ S' = 702 ----(2)

Now, we were to find sum of all 2-digit positive numbers which when divided by 7 yield 2 or 5 as remainder.

So, we will add both (1) and (2).

So,

⇒ Sum = S + S'

⇒ Sum = 654 + 702

⇒ Sum = 1356

Therefore, sum of all 2-digit positive numbers which when divided by 7 yield 2 or 5 as remainder is 1356.

Answer 2

*Question:—

The Sum of all 2 digit positive numbers which when divided by 7 yeild 2 or 5 as remainder is ...

*Answer :—

Formula used :—

n

∑ = 1/2n ( n + 1 )

k=1

13

∑ (7r+2) = 7 × 2 + 13/ 2 × 6 + 2 × 12

r=2

= 7 × 90 + 24 = 654

13

∑ (7r+5) = 7 ( 1 + 13/2 ) × 13 + 5 × 13 = 702

r=1

Total = 654 + 702 = 1356.

So we get the answer 1356 as remainder