Concept: Remainders

Remainder means something which is ‘left over’ or ‘remaining’.

If you have 11 chocolates and you share it equally amongst your four friends. How many chocolates will remain not shared?
If you give two chocolates each to your friends, you would have shared 8 chocolates. 3 chocolates will remain with you, and this leftover of 3 chocolates is called the remainder.

Mathematically we can write the above expression as:
11 ÷ 4 = 2 remainder 3 or
11 = 4 × 2 + 3
Where 11 is the dividend, 4 is the divisor, 2 is the quotient, and 3 is the remainder.

On dividing 25 by 7. We get 3 equal parts of 7 each, that add up to 21
3 x 7 = 21
We are left with 4. This 4 is the remainder.
⇒ 25 = 7 × 3 + 4

 

$\mathrm{R}\left[\frac{{\mathrm{N}}_1+{\mathrm{N}}_2}{\mathrm{d}}\right]$ = $\mathrm{R}\left[\frac{{\mathrm{N}}_1}{\mathrm{d}}\right]$ + $\mathrm{R}\left[\frac{{\mathrm{N}}_2}{\mathrm{d}}\right]$

$\mathrm{R}\left[\frac{{\mathrm{N}}_1\times {\mathrm{N}}_2}{\mathrm{d}}\right]$ = $\mathrm{R}\left[\frac{{\mathrm{N}}_1}{\mathrm{d}}\right]$ × $\mathrm{R}\left[\frac{{\mathrm{N}}_2}{\mathrm{d}}\right]$

Note: In case sum / product of remainders is greater than or equal to the divisor, then we can get the final remainder by again dividing by the divisor.

 

 

When N is divided by d, remainder will be any integer from 0 till (d - 1).

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: When 14 is divided by 5, remainder is 4, while the negative remainder is (4 - 5) = -1.

As discussed earlier, when N objects are divided in groups of d leaving remainder r, it means there are r objects extra. It also means that to make another complete group of d objects we need to (d - r) more objects i.e., we are short of (d - r) objects.

 

Fermat's Little Theorem states that remainder of a^p-1 when divided by p leaves a remainder of 1 if p is a prime number and a & p are co-prime numbers.

i.e., $\mathrm{R}\left[\frac{{\mathrm{a}}^{\mathrm{p}-1}}{\mathrm{p}}\right]$ = 1

Note: Questions based on Fermat's Theorem have not been asked in CAT in recent few years.

 

According to Euler’s theorem, any number a raised to the power Φ(d) will leave a remainder of 1 when it is divided by d, provided d and a are co-primes.

$\mathrm{R}\left[\frac{{\mathrm{a}}^{\mathrm{\Phi }\left(\mathrm{d}\right)}}{\mathrm{d}}\right]$ = 1

Φ(d) = number of coprimes to d which are less than d.

Note: Questions based on Euler's thorem have not been asked in CAT in recent few years.

 

When (p - 1)! is divided by p, the remainder will be (p – 1), where p is a prime number.

$\mathrm{R}\left[\frac{(\mathrm{p}-1)!}{\mathrm{p}}\right]$ = p - 1

Example: Remainder of 16! when divided by 17 is 16.

Based on Wilson's theorem we get another rule:

When (p - 2)! is divided by p, the remainder will be 1, where p is a prime number.

$\mathrm{R}\left[\frac{(\mathrm{p}-2)!}{\mathrm{p}}\right]$ = 1

Proof:
$\mathrm{R}\left[\frac{(\mathrm{p}-1)!}{\mathrm{p}}\right]$ = p - 1

⇒ $\mathrm{R}\left[\frac{(\mathrm{p}-1)}{\mathrm{p}}\right]$ × $\mathrm{R}\left[\frac{(\mathrm{p}-2)!}{\mathrm{p}}\right]$ = (p - 1)

⇒ (p - 1) × $\mathrm{R}\left[\frac{(\mathrm{p}-2)!}{\mathrm{p}}\right]$ = (p - 1)

∴ $\mathrm{R}\left[\frac{(\mathrm{p}-2)!}{\mathrm{p}}\right]$ = 1

Note: Questions based on Wilson's thorem have not been asked in CAT in recent few years.