Number Systems: Finding Remainders

# Number Systems: Finding Remainders | General Aptitude for GATE - Mechanical Engineering PDF Download

 Table of contents What is a Remainder? Finding Remainders of a Product (Derivative of Remainder Theorem) Finding Remainders Of Powers With The Help Of Remainder Theorem Application Of Binomial Theorem In Finding Remainders Wilson's Theorem Fermat's Theorem Important Points

## What is a Remainder?

• When you divide one number, called the "dividend," by another number, known as the "divisor," the result is expressed as a fraction, like "dividend/divisor." In a simple example, such as dividing 6 by 3 (6/3), the answer is 2, which is the "quotient." However, not all division problems are straightforward like 6/3; some result in remainders.
• To put it simply, a remainder is the fractional part left over when you divide two numbers, and the division doesn't result in a whole number quotient. For example, when you divide 8 by 3, the remainder is 2. This means you can have two sets of 3, and there will be 2 left over.

• Thinking of remainders as mixed numbers can be helpful. For instance, the fraction 8/3 is equivalent to the mixed number 2 . Here, 2/3 represents the remainder, indicating that 2 parts are left out of the 3 parts needed to make a whole number. The denominator of the fraction will always be the same as the divisor.
• In more complex problems, such as dividing a dividend by a divisor to get a whole number quotient and a remainder, the relationship is expressed by the equation: dividend/divisor = quotient + remainder/divisor. For example, in the division problem 8/3, where 8 is the dividend, 3 is the divisor, 2 is the quotient, and 2 is the remainder, this equation holds true.

## Finding Remainders of a Product (Derivative of Remainder Theorem)

(i) If ‘a1’ is divided by ‘n’, the remainder is ‘r1’ and if ‘a2’ is divided by ‘n’, the remainder is r2. Then,

If a1+a2 is divided by n, the remainder will be r1 + r2.
If a1 - a2 is divided by n, the remainder will be r1 - r2.
If a1 × a2 is divided by n, the remainder will be r1 × r2.

### Concept of Negative Remainder

By definition, remainder cannot be negative. But in certain cases, you can assume that for your convenience. But a negative remainder in real sense means that you need to add the divisor in the negative remainder to find the real remainder.

Example:

If 21 is divided by 5, the remainder is 1 and if 12 is divided by 5, the remainder is 2. Then, if (21 + 12 = 33) is divided by 5, the remainder will be (1 + 2 = 3).
If (21 - 12 = 9) is divided by 5, the remainder will be 1 - 2 = - 1.
But if the divisor is 5, - 1 is nothing but 4 (9 = 5 × 1 + 4)
So, if 9 is divided by 5, the remainder is 4 and 9 can be written as 9 = 5 × 2 - 1.
So here - 1 is the remainder. So - 1 is equivalent to 4 if the divisor is 5. Similarly - 2 is equivalent to 3.
If (21 × 12 = 252) is divisible by 5, the remainder will be (1 × 2 = 2).

(ii) If two numbers ‘a1’ and ‘a2‘ are exactly divisible by n. Then their sum, difference and product is also exactly divisible by n.
i.e., If ‘a1’ and ‘a2’ are divisible by n, then

a1 + a2 is also divisible by n
a1 - a2 is also divisible by n
a1 × a2 is also divisible by n.

Example: 12 is divisible by 3 and 21 is also divisible by 3
Sol. So, 12 + 21 = 33, 12 - 21 = - 9 and 12 × 21 = 252 all are divisible by 3.

## Finding Remainders Of Powers With The Help Of Remainder Theorem

Example 1: What is the remainder if 725 is divided by 6?
Solution: If 7 is divided by 6, the remainder is 1. So if 725 is divided by 6, the remainder is 1 (because 725 = 7 × 7 × 7… 25 times. So remainder = 1 × 1 × 1…. 25 times = 125).

Example 2: What is the remainder, if 363 is divided by 14.
Solution: If 33 is divided by 14, the remainder is - 1. So 363 can be written as (33)21.
So the remainder is (- 1)21 = - 1.
If the divisor is 14, the remainder - 1 means 13. (14 - 1 = 13) by pattern method.

Question for Number Systems: Finding Remainders
Try yourself:Find the remainder when 433 is divided by 7.

## Application Of Binomial Theorem In Finding Remainders

The binomial expansion of any expression of the form
(a + b)n = nCo an + nC1 an-1 × b1 + nC2 × an-2 × b2 ..... + nCn-1 × a1 × bn-1 + nCn × bn
Where nCo, nC1, nC2, .... are all called the binomial coefficients
In general, nCr = n!/r!(n - r)!

There are some fundamental conclusions that are helpful if remembered:
(a) There are (n + 1) terms.
(b) The first term of the expansion has only a.
(c) The last term of the expansion has only b.
(d) All the other (n - 1) terms contain both a and b.
(e) If (a + b)n is divided by a, then the remainder will be bn such that bn < a.

Example : What is the remainder if 725 is divided by 6?
Solution.
(7)25 can be written (6 + 1)25. So, in the binomial expansion, all the first 25 terms will have 6 in it. The 26th term is (1)25. Hence, the expansion can be written 6x + 1. 6x denotes the sum of all the first 25 terms.

Since each of them is divisible by 6, their sum is also divisible by 6, and therefore, can be written 6x, where x is any natural number. So, 6x + 1 when divided by 6 leaves the remainder 1.

(OR)

When 7 divided by 6, the remainder is 1. So when 725 is divided by 6, the remainder will be 125 = 1.

Question for Number Systems: Finding Remainders
Try yourself:Remainder when 2510 is divided by 576?

## Wilson's Theorem

• If n is a prime number, (n - 1)! + 1 is divisible by n.
Lets take n = 5
Then (n - 1)! + 1 = 4! + 1 = 24 + 1 = 25 which is divisible by 5.
Similarly If n = 7
(n - 1)! + 1 = 6! + 1 = 720 + 1 = 721 which is divisible by 7.
Corollary
If (2p + 1) is a prime number (p!)2 + (- 1)p is divisible by 2p + 1.

For Example
If p = 3, 2p + 1 = 7 is a prime number
(p!)+ (- 1)p = (3!)2 + (- 1)3 = 36 - 1 = 35 is divisible by (2p + 1) = 7.
• Property
If “a” is natural number and P is prime number then (ap - a) is divisible by P.
Example: If 231 is divided by 31 what is the remainder?

So remainder = 2

## Fermat's Theorem

• If p is a prime number and N is prime to p, then Np -1 - 1 is a multiple of p.
• Corollary
Since p is prime, p - 1 is an even number except when p = 2.
Therefore ( ) = M(p).
Hence either  -1 is a multiple of p, that is  = Kp ± 1, where, K is some positive integer.

## Important Points

• The sum of consecutive five whole numbers is always divisible by 5.
• The square of any odd number when divided by 8 will leave 1 as the remainder
• The product of any three consecutive natural numbers is divisible by 6.
• The unit digit of the product of any nine consecutive numbers is always zero.
• For any natural number n, 10n-7 is divisible by 3.
• Any three-digit number having all the digits same will always be divisible by 37.
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## FAQs on Number Systems: Finding Remainders - General Aptitude for GATE - Mechanical Engineering

 1. How can we find the remainder of a product using the Derivative of Remainder Theorem?
Ans. The Derivative of Remainder Theorem states that if we divide a polynomial by a linear polynomial, the remainder is the same as the value of the polynomial at the root of the linear polynomial.
 2. How can we find the remainder of powers with the help of the Remainder Theorem?
Ans. To find the remainder of powers using the Remainder Theorem, we first express the power as a polynomial. Then, we divide the polynomial by the given divisor and find the remainder.
 3. How can we apply the Binomial Theorem to find remainders?
Ans. We can apply the Binomial Theorem to find remainders by expanding the given expression using the Binomial Theorem and then dividing the resulting polynomial by the given divisor to find the remainder.
 4. What is Wilson's Theorem?
Ans. Wilson's Theorem states that for any prime number p, (p-1)! is congruent to -1 (mod p). This theorem is useful in number theory and finding remainders in various mathematical problems.
 5. What is Fermat's Theorem?
Ans. Fermat's Theorem, also known as Fermat's Little Theorem, states that if p is a prime number and a is any integer not divisible by p, then a^(p-1) is congruent to 1 (mod p). This theorem is helpful in finding remainders in number theory and cryptography.

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