Question

8. It is possible to pair up all the numbers from 1 to 70 so that the positive difference of the numbers in each pair is always the same. For example, one such pairing up is (1,2), (3,4), (5,6),….(69,70). Here the common on difference is 1. What is the sum of all such common differences.

Answer 1

Hi there!
Here's the answer:

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Given,
Range = 1 - 70

Condition :
In two numbers, positive difference of the numbers in each pair should be same.

Here,
Two things have to be taken into account:
• Common difference
• No. of such pairs

For simplicity,
Consider range as 1 - 10
Here, when the common difference exceeds 5, numbers are not paired as per the condition.

Like if you take 6 as common difference, not all the pairs will have same value, because it is impossible.

•°• For range 1 - n, we have (n/2) common differences.

Now,
Coming to the given range 1 - 70
n = 70
=> (n/2) = 35

•°• No. of common differences that will satisfy the condition = 35

Now, Find sum of these differences.
1 + 2 + 3 + …… + 35
= (35 × 36) ÷ 2
= 630


•°• Required sum of all such common differences = 630


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:)


Hope it helps

Answer 2

Answer:

total numbers( 1 to 70)=70

if we want to make pair of 70 numbers ..i.e= 70/2= 35.

Now we need to find out total possible common difference for all pairs i.e (Or we can say that all divisor of 35)= 1,5,7 and 35.

Sum of all common differences = (sum of all common divisor)+( no. of pairs)

                                         =(1+5+7+35)+(35)

                                =48*35= 1680. Ans

Step-by-step explanation: