Number System 6

BASE and INDEX

If a number ‘b’ is multiplied by it self ‘n’ times, then the product is called the nth power of b, i.e.

b x b x b x b ……. Up to n times = bⁿ

Here, b is called base and n is called the index.

Last digit ( Digit at unit place) in (xyz)ⁿ

Here the given number is (xyz)ⁿ, ‘xyz’ is base and ‘n’ is the index and ‘z’ is the last digit of the base.

To find out the last digit in (xyz)ⁿ, following steps are to be followed.

Divide the index (n) by 4, then

CASE – 1

If remainder = 0, then check if ‘z’ is odd (except 5), then last digit = 1.

And if ‘z’ is even, then last digit = 6.

CASE - 2

If remainder = 1, then required last digit = last digit of base (i.e. ‘z’)

If remainder = 2, then required last digit = last digit of (z)²

If remainder = 3, then required last digit = last digit of (z)³

Note:- If z is 5, then last digit in the product = 5.

Ex.:- Find the last digit in (295073)¹³⁰

Soln.:- Dividing 130 by 4, the remainder = 2

Referring to Case - 2, the required last digit is the last digit of (z)², i.e. (3)² = 9

Number of Zeroes at the end of product

On multiplying two or more given numbers, the zeros are produced at the end of the resulting product due to the following reasons:

(a) If there is any zero at the end of the factors (or numbers being multiplied)

Ex.:- 7 x 20 =140, the zero of 20 is produce at the end of the product also.

(a) If 5 or a multiple of 5 is multiplied by any even number.

Ex.:- 45 x 12 = 540, multiple of 5 is multiplied by even number 2

Combining the above two reasons, we may say that:

(i) Resolve all the given number into their factors.

(ii) Count the number of 2s and 5s. i.e. (5)^x x (2)^y.

(iii) No. of zeroes at the end of the product = No. of the 2s or no. of 5s, whichever is less