Question

If k is an integer such that 55 < k < 70, what is the value of k ? 
Statement 1:- K can be written as a sum of 8 consecutive numbers. 
Statement 2:- If k were divided by 3, the remainder would be 0.​

Answer 1

Reasoning:

We can find out from statement 1:-

• k is a multiple of 4.

Now, k is both a multiple of 3 and 4.

Hence it is a multiple of 12, between 55 and 70.

Possible number 60, is in the range.

Now, the integer k must be 60.

Reason:

The consecutive integers can be shown as the below ones.

• $\sf{x,\; x+1,\; x+2,\; x+3,\; x+4,\; x+5,\; x+6,\; x+7}$

Their sum:

$\sf{=8x+(0+1+...+6+7)}$

$\sf{=8x+(0+7)\times \dfrac{8}{2} }$ [Little Gauss Trick]

$\sf{=8x+28}$

$\sf{=4(2x+7)}$

$\sf{4(2x+7)}$ is necessarily a multiple of 4.

For more:

Little Gauss trick, how did Gauss add from 1 to 100?

Gauss found out that the first and last sum is always 101.

There are 100 numbers

Now, since we add 2 numbers at once, there will be 50 times of addition.

$\sf{101\times 50=5050}$ was Gauss's answer to the teacher's question.

The teacher and all the students were astonished. He was 8 years old.

Answer 2

Given : k is an integer such that 55 <k< 70,

Statement 1   :-K can be written as a sum of 8 consecutive numbers.

Statement 2 -If k were divided by 3, the remainder would be 0

To Find :  the correct option

1.the question can be answered by using one of the statements alone, but cannot be answered by using the other statements alone.

2. the question can be answered by using both statements together, but cannot be answered by using either statement alone.

3.the question can be answered by using either statement alone.

4.Questions cannot be answered even by using both the statements together.​

Solution:

k is an integer such that 55 <k< 70,

K is from 56  to  69

Statement 1:-K can be written as a sum of 8 consecutive numbers.

Let say 8 consecutive numbers are

n , n+ 1 ,......................., n + 7

Then sum = 8n + 1 + 2 +................+ 7

= 8n + 7 (8)/2

= 8n + 28

55  < 8n + 28   < 70

=> 27 <  8n  <  42

=>  4 ≤ n ≤ 5

8n + 28  

60  , 68

( not unique answer)

Statement 2 -If k were divided by 3, the remainder would be 0

Then possible values are 57 , 60 ,63 , 66

( not unique answer)

Statement 1 :   60  , 68

Statement 2  :  57 , 60 ,63 , 66

From Both 60

Value of K = 60

the question can be answered by using both statements together, but cannot be answered by using either statement alone.

Learn More:

k is an integer such that 55 <k< 70

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