10. Find the digit at the unit's
place of
(377)59 x (793)87 x (578)129 x
(99)99
(a) I (b) 2.
(c) 7
(d) 9
Answers
Answer:
It's too simple dear.... It's all about binomial theorem....
Let me send you the solution....
Wait.....
So I think (b) is the correct answer... Sorry for being late I thought it will not be easy to do it binomially as it can be hard to understand...
Simple trick is that whenever you multiply two numbers the last digit or unit place is simply the multiplication of last digits of given two numbers...
For example..
5892*2569 = 15136548...
Last digit is 8...
So for 5892 it's 2 at unit place... And for 2569 it is 9
So 2*9=18...and therefore last digit is 8....
Any doubts?....
Step-by-step explanation:
(377)59 × (793)87 × (578)129 × (99)99Number System l 31
6 5 4 3 2 1 1 2 3 4 5 60
Each point on the number line represents a unique real number and
each real number is denoted by a unique point on the number line.
Symbols of some special sets are:
N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
R : the set of all real numbers
Z+ : the set of positive integers
Q+ : the set of positive rational numbers, and
R+ : the set of positive real numbers
The symbols for the special sets given above will be referred
to throughout the text.
Even Integers
An integer divisible by 2 is called an even integer. Thus, ..., – 6, –
4, – 2, 0, 2, 4, 6, 8, 10, 12,...., etc. are all even integers. 2n always
represents an even number, where n is an integer.
For example, by putting n = 5 and 8 in 2n, we get even integer
2n as 10 and 16 respectively.
Odd Integers
An integer not divisible by 2 is called an odd integer.
Thus, ..., –5, –3, –1, 1, 3, 5, 7, 9, 11, 13, 15,..., etc. are all
odd integers.
(2n – 1) or (2n + 1) always represents an odd number, where
n is an integer.
For example by putting n = 0, 1 and 5 in (2n – 1), we get odd
integer (2n – 1) as – 1, 1 and 9 respectively.
Properties of Positive and Negative Numbers
If n is a natural number then
(A positive number)natural number = A positive number
(A negative number)even positive number = A positive number
(A negative number)odd positive number = A negative number
CONVERSION OF RATIONAL NUMBER OF
THE FORM NON-TERMINATING RECURRING
DECIMAL INTO THE RATIONAL NUMBER OF
THE FORM p
q
First write the non-terminating repeating decimal number in
recurring form i.e., write 64.20132132132.
................................ as 64 20132.
Then using formula given below we find the required p
q
form
of the given number.
Rational number in the form p
q
=
Complete number neglecting
the decimal and bar over
repeating digit( )
Non- recurring part of
the number neglecting
the decs
−
imal
m n times 9 followed by times 0
where m = number of recurring digits in decimal part
and n = number of non-recurring digits in decimals part
Thus, p
q
form of 64 20132. = 6420132 6420
99900
−
= 6413712
99900
534476
8325 =
In short; 0.a = , 0. , 0. , etc.
9 99 999
a ab abc ab abc = =and
0.ab = , 0. , 0. , 90 990 900
ab a abc a abc ab abc abc − − − = =
0. , 9900
abcd ab abcd − = , etc. 990
abcde abc ab cde − ⋅ =
Illustration 1: Convert 2 46102 . in the p
q
form of rational
number.
Solution: Required p
q
form =
246102 2
99999
246100
99999
− =
Illustration 2: Convert 0 1673206 . in the p
q
form of
rational number.
Solution: Required p
q
form =
1673206 167
9999000
1673039
9999000
− =
Illustration 3: Convert 31.026415555 ... into p
q
form of ra-
tional number.
Solution: First write 31.026415555... as 31 026415.
Now required p
q
form =
31026415 3102641
900000
27923774
900000
− =
= 13961887
450000 .
DIVISION
4 275 68
24
35
32
3
Here 4 is the divisor, 275 is the dividend,
68 is the
Remainder is always less than divisor.
quotient and 3 is the remainder.
Thus, Divisor Quotient Dividend
abc
Remainder
Thus,
Dividend = Divisor × Quotient + Remainder
For example, 275 = 4 × 68 + 3
When quotient is a whole number and remainder is zero, then
dividend is divisible by divisor.
TESTS OF DIVISIBILITY
I. Divisibility by 2:
A number is divisible by 2 if its unit digit is any of 0, 2,
4, 6, 8.
Ex. 58694 is divisible by 2, while 86945 is not divisible
by 2.
II. Divisible by 3:
A number is divisible by 3 only when the sum of its digits
is divisib