Aptitude - Numbers - Discussion

Please give me answer in more better way?

Check numbers from 3, 4, 5, ... , in the above given format. We will get remainder 1 less than the number, so for 67^67+67 = 66.

(67^67+67)/68.

--> (67*67*67*...67)/68+(67)/68.

--> 67-68 = -1.

--> (-1*-1*-1*....*-1)/68+(67)/68.

--> (-1/68)+67/68.

--> 68-1= 67.

--> (67+67)/68 = 2.

68-2 = 66.

Nothing complexity with the solution.

From the formula : (x^n+1) is always divisible by (x+1) only when n is odd.

In the above problem n is 67 i.e, n is odd.

So, we can write (67^67+67) as ((67 ^ 67)+1) + 66.

Why because making the given expression as convenient to us.

Thank you.

Is there any Short trick?

Just use remainder theorem see how,

Write down the given expression in the form of polynomial of x :67^67+67 = x^67+x, now by remainder theorem (x-(-1)) i.e. 68,

So x^67+x /(x-(-1)) will give you remainder as -2, add this to 68 to get answer 66.

67^67 = (68^67-1^67).

So (68^67)+67-1^67.

= (68^67)+66.

= (68^67) is completely divisible by 68. So remainder is 66.

It's quite simple. Find unit digit of power value add it with their next value. Divide with divisor. Then subtract remainder from original divisor.

For ex: (3^3+3)/4.

Unit digit of 3^3 is 7. Add with 3 as it is given 7+3=10.

Divide 10/4 remainder=2.

Now finally subtract 2 with 4 so 4-2=2.

Let's see this one: (67^67+67)/68.

Unit digit of 67^67 is 3 as (7^16*7^3). Add this with 67.67+3 = 70.

Divide 70/68 = remainder = 2.

Finally subtract 2 with 68 so 68-2 = 66 answer.

Hi guys,

Here you can use concept of negative reminders.

68/67 here positive reminder as all we know 68, but negative reminder is -1(67-68).

For example 11/3 positive reminder 2 and negative reminder -1(11-12).

Here is our question.

Rem of (67^67+67)/68 = (-1)^67+(-1) = -2.

So the answer is 68-2 = 66.

Hi guys,

Here you can use concept of negative reminders.

68/67 here positive reminder as all we know 68, but negative reminder is -1(67-68).

For example 11/3 positive reminder 2 and negative reminder -1(11-12).

Here is our question.

Rem of (67^67+67)/68 = (-1)^67+(-1) = -2.

So the answer is 68-2 = 66.