Number System is an important concept for solving GATE Aptitude questions and aptitude for entrance exams for different companies.

The below is an important question which has been asked in many exams.

**Question :**

In 7^{126} is divisible by 48, find reminder?

**Normal Approach :**

For calculating the reminder we first calculate the original value for number 7^{126} and divide it further to and obtain the reminder.

It is very long and time taking process and it is not at all feasible to solve it in this way. So we use some important mathematical concepts related to divisibility to solve this problem.

**Speedy approach :**

Important concepts for solving problem,

- (x
^{n}– a^{n}) divisible by (x – a) for every n (n belongs to integers)- (x
^{n}– a^{n}) divisible by (x + a) for every even number n (n belongs to integers)- (x
^{n}– a^{n}) divisible by (x + a) for every odd number n (n belongs to integers)

And we also use another basic formula;

Dividend = divisor x quotient + reminder

The given number in form such that the base is very near to 48.

This is done by using the formula ( a^{mn }) = (a^{m})^{n}

7^{126}= (7^{2})^{63}= 49^{63}

Now by using our mathematical formulae we should add or subtract a number to 49^{63} such that it is divisible by 48.

(49^{63}– 1) = (49^{63}- 1^{63})

By comparing it with (x^{n} – a^{n}) we can write,

x = 49, n = 63 and a = 1

Therefore from the above we get that

( 49^{63} – 1^{63}) is divisible by (49-1) and (49+1)

So, (49^{63} – 1) is divisible by 48

Let (49^{63} – 1)/48 = q (where q is the quotient )

49^{63}– 1 = 48 x q 49^{63}= 48 x q + 1 7^{126}= 48 x q + 1

Comparing with.

Dividend = divisor * quotient + reminder

So from above when Dividend = 7^{126}, divisor = 48 then reminder is 1.

So when 7^{126} is divided by 48 the reminder is 1.

In this way we can obtain the reminder for such large numbers. It takes very less time and very useful in competitive exams.

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