05. Aptitude For C.A.T.-- Numbers - Theory - Remainder Theorem

Numbers - Important Concept and Formulae

Finding the number of zeroes at the end of any factorial:

How to find the number of zeroes at the end of the product of numbers

The number of zeroes at the end of the product depends upon the number of 2‘s and 5’s in the expression.

The number of zeroes is the same as the number of pair of (5 x 2) units.

Since the number of two’s will be more than the number of fives

So, we have to count the number of five’s to get the number of zeroes at the end.

Example : Find the number of zeroes in 10!?

Solution : 10!= 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1.

Since the number of 2 is more than the number of 5’s, so the number of 5’s will be counted to form the combination of 2 x 5.

We have two 5’s, one digit 4 and another 5 is 10 ( since 10= 5 x2)

Thus, there are only 2 zeroes at the end of 10!

Example: Find the number of zeroes at the of the product of 2¹¹¹ x  5²²²

Solution : In the given product, the number of 2’s are less than the number of 5’s, hence we take the number of 2’s into consideration.

Thus there will be only 111 pairs of(2 x5).

Therefore, the number of zeroes at the end of the product of the given expression will be 111.

Finding the largest power of a number contained in a factorial:

How to find the highest power of x that can exactly divide n! : We divide n by x, n by x² … and so on till we get $\frac{n}{x^{k}}$ equal to 1.

Example : Find the largest power of 5 contained in 120!?

Solution : [$\frac{120}{5}$ ] + [$ \frac{120}{5^{2}}$ ] = 24 + 4  =28

We need not further, because 120 is not divisible by 5³ (=125)

Example 2: Find the largest power of 7 that can exactly divide 800!?

Solution : $\frac{800}{7}$   + $\frac{800}{7^{2}}$   + $\frac{800}{7^{3}}$

  = 114 + 16 +2 = 132

Thus the highest power of 7 is 132 by which 800! can be divided.

Concept of Remainder :

Definition of Remainder: The part of the dividend that is not evenly divided by the divisor, is called the remainder.

Rule 1 :

When x1, x2, x3 … are divided by n individually, the respective remainders obtained are  R1, R2, and R3 etc .. when ( x1 + x2 + x3) is divided by n the remainders can be obtained by (R1+ R2+R3+ ..) by n.

Example : Find the remainder when 65+ 75+ 87 is divided by 7?

Solution: When we divided 65, 75 and 87, we obtain the respective remainders as 2, 5 and 3.

Now we can get the required remainder by dividing the sum of the remainders by 7 = $\frac{(2 + 5 + 3)}{7}$ =3

Instead of dividing the sum of original numbers (65 +75+87 =227) by 7, which will also give the same remainder 3.

Example: Find the remainder when 719 x 121 x 237 x 725 x 727 is divided by 14 ?

Solution :
First, we find the remainders when each individual number is divided by 14.

The remainder when 719 is divided by 14 is 5

The remainder when 121 is divided by 14 is 9

The remainder when 237 is divided by 14 is 13

The remainder when 725 is divided by 14 is 11

The remainder when 727 is divided by 14 is 13

Now we find the remainder of the remainders.

 Hence, the remainder of $\frac{ 719 \,\times\, 121 \,\times\, 237 \,\times\, 725 \,\times\, 727}{14}$        
       =$\frac{5\,\times\, 9\,\times\, 13\,\times\, 11\,\times\, 13}{14}$

ð  $\frac{45 \,\times\, 13 \,\times\, 11 \,\times\, 13}{14} $

ð  $ \frac{3 \,\times\, 13 \,\times\, 11 \,\times\, 13}{14}$

ð  $\frac{39 \,\times\, 143}{14}$

ð  $\frac{11 \,\times\, 3}{4}$

ð  $\frac{33}{4}$

ð  5

Example: What is the remainder when 7⁹ is divided by 4?

Solution :  $\frac{7^{9}}{4}$ = $ \frac{7\,\times\, 7 \,\times\, 7 \,\times\, 7 \,\times\, 7\,\times\, 7 \,\times\, 7 \,\times\, 7 \,\times\, 7}{4} $

                          = $\frac{3 \,\times 3 \,\times\, 3 \,\times\, 3\,\times\, 3 \,\times\, 3 \,\times\, 3 \,\times\, 3 \,\times\, 3}{4} $

                         = $ \frac{27 \,\times\, 27 \,\times\, 27}{4}$

                         = $\frac{3 \,\times\, 3 \,\times\, 3}{4}$

                         =4

             Thus the required remainder is 4.

Rule 2 :

A certain number when successively divided by two different numbers leaves some remainder. The same number, when divided by the product of the two divisors, will leave remainder equal to :
(First Divisor x  Second Remainder )  + First Remainder.

Example: A certain number when successively divided by 5 and 9 leaves the remainders 2 and 7 respectively. What is the remainder when the same number is divided by 45?

Solution :
 Remainder =( First Divisor x Second Remainder ) +  First remainder

                               = (5 x 7) +2 =37

Rule 3 :

If two numbers M and N when separately divided by x, leaving the remainders a and b respectively. Then M^p  x N^q,  when divided by x, will leave the remainder a^m  x bⁿ

Example : Find the remainder when 80² x 53³ is divided by 26.

Solution :80 and 53, when divided by 26, leave remainders 4 and 1 respectively.

  Therefore, 80² x 53³ when  divided by 26 leaves remainder  4² x  1³ =16

Rule 4 :

Finding whether (aⁿ – k) is divisible by x?

Method :

1. aⁿ is divided into some parts in such a way that the value of each part is slightly more or less than the multiple of x

2. Take remainder when each part is divided by x

3. Multiply all the remainders obtained from each part

4. Subtract k from the product of remainders

5. If the result is a multiple of x or zero, then (aⁿ – k) is exactly divisible by y.

Example: Is 7⁸ -1 divisible by 24?

Solution : 7⁸ -1 = 7² x 7² x7² x7²  -1= 49 x 49 x 49 x 49 – 1

           The remainder we get when each part 49 is divided by 24 = 1 x 1 x 1 x 1 -1 =0

     Therefore, 7⁸ -1 divisible by 24

Concept of Negative Remainders:

Remainders are always non-negative by their definition.

When 14 is divided by 5, the usual remainder would be 4. But it can also be expressed as -1. It is a negative remainder. The concept of negative remainders sometimes helps the student in reducing calculations.

For numbers like 12,19,26,33 etc., we can represent them as 7k+5 or 7k-5.

Example: Find the remainder when 262 x 68 is divided by 18?

Thus , $\frac{262 \,\times\, 68 }{18}$ will give remainder $\frac{-10 \,\times\, -4 }{18}$ = $\frac{40 }{18}$ = 14

This will give 14 as remainder.

So, using the concept of negative remainders, we can reduce our calculations

Example: What is the remainder when 11²¹⁵ is divided by 12?

Solution: The remainder we get when 11  divided by 12  is 11 = -1

The negative remainder is -1.

   So 11²¹⁵ =(-1)²¹⁵ = -1 =11

Using the concept of the negative remainder, we can say the remainder is 11.