The Lightning Fast Addition!

A  story tells that, as a 10-year-old schoolboy, Carl Friedrich Gauss was asked to find the sum of the first 100 integers. The tyrannical schoolmaster, who had intended this task to occupy the boy for some time, was astonished when Gauss presented the correct answer, 5050, almost immediately.

How did Gauss find it?

Actually, he used this trick! 

 

Trick for The Lightneing Fast Addition!

Why lightning fast speed needed?

Gauss attached 0 to the series and made pairs of numbers having addition 100.

100 + 0 = 100

99 + 1 = 100

98 + 2 = 100

97 + 3 = 100

96 + 4 = 100

95 + 5 = 100
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..
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51 + 49 = 100

This way he got 50 pair of integers (ranging in between 1-100) having sum equal to 100.

So sum of these 50 pairs = 100x50 = 5000.

 
And the number 50 left added to above total to get sum of integers 1 - 100 as 5000 + 50 = 5050

What Could Be The Product?

Zach chooses five numbers from the set {1, 2, 3, 4, 5, 6, 7} and tells their product to Claudia. She finds that this is not enough information to tell whether the sum of Zach’s numbers is even or odd. What is the product that Zach tells Claudia?

Here is the way to get that product!

Guessing The Correct Product in Question!

What was the question?

When Zach tells the product of 5 numbers that he has chosen he indirectly conveying product of 2 un chosen numbers.

For example, product of all numbers in set = 1 x 2 x 3 x 4 x 5 x 6 x 7 = 5040 and if Zach tell product 1 x 2 x 4 x 5 x 7 = 280 then obviously the product of numbers that he hasn't chosen is 5040/280 = 18 = 3 x 6.

Now, there are only 2 products viz. 12 and 6 which have more than 1 pair of numbers. 

The product 12 can be from pairs - (3,4) and (6,2)

The product 6 can be from pairs - (1,6) and (2,3) 

Here if the sum of un chosen numbers is odd (or even) then sum of other 5 chosen numbers also must be odd (or even).

In above cases, 6 has pairs whose sum is odd always and hence sum of other 5 numbers would be odd. In that case, Claudia would have been sure with if sum of numbers selected by Zach is either odd or even.

While in other case, the product 12 has 1 pair having sum odd (3,4) and other pair having sum even (6,2). Hence, Zach must have 'indirectly' suggested product 12 as a product of un chosen numbers that's why Claudia is saying that she doesn't know if the sum of numbers selected by Zach is even or odd.

Hence, the product of numbers selected by Zach = 5040 / 12 = 420.