NUMBER SYSTEM
- Number System is a method of representing Numbers on the Number Line with the help of a set of Symbols and rules.
- These symbols range from 0-9 and are termed digits.
- Number Tree:
The number of zeros:
It is nothing else but the number of zeros at the end i.e. Number of trailing zeros. Just to make you clear, 170130000 has 5 zeros but 4 trailing / ending zeros.
The number of trailing zeros in a Product or an Expression:
If we look at a number N, such that N = 2580000 = 258 x 104
The number of trailing zeros is the Power of 10 in the expression or in other words, the number of times N is divisible by 10.
We know 10 = 2 x 5
For a number to be divisible by 10, it should be divisible by 2 & 5, since making a pair of 2 and 5 will give us 10.
So, the number of trailing zeros is going to be the power of 2 or 5, whichever is less.
The number of trailing zeros in Power of an expression:
We know that
101 = 10........ one trailing zero
102 = 100........ two trailing zeros
103 = 1000....... three trailing zeros
1002 = 10000 …… four trailing zeros
Thus for finding the number of zeros in Np If N has t trailing zeros
Then Np has t x p trailing zeros
NUMBER OF FACTORS OF A NUMBER:
If N is a composite number such that N = apbqcr …… where a, b, c are prime factors of N and p, q, r …… are positive integers, then the number of factors of N is given by the expression.
= (p + 1) (q + 1) (r + 1) …...
Example: Find the number of factors of 140
Solution:
140 = 22 × 51 × 71
Hence 140 has (2 + 1) (1 + 1) (1 + 1), i.e., 12 factors.
Please note that the figure arrived at by using the above formula includes 1 and the given number N also as factors. So, if you want to find the number of factors the given number has excluding 1 and the number itself, we find out (p + 1) (q + 1) (r + 1) and then subtract 2 from that figure.
In the above example, the number 140 has 10 factors excluding 1 and 140 itself.
DIVISIBILITY RULES:
Divisibility by 1: Any integer (not a fraction) is divisible by 1.
Divisibility by 2: The last digit is even (0,2,4,6,8)
Divisibility by 3: The sum of the digits is divisible by 3
Divisibility by 4: The last 2 digits are divisible by 4
Divisibility by 5: The last digit is 0 or 5
Divisibility by 6: Is even and is divisible by 3 (it passes both the 2 rule and 3 rule above)
Divisibility by 8: The last three digits are divisible by 8
Divisibility by 9: The sum of the digits is divisible by 9
Divisibility by 10: The number ends in 0
Divisibility by 11: Add and subtract digits in an alternating pattern (add a digit, subtract next digit, add next digit, etc). Then check if that answer is divisible by 11.
Unit Digit:
The unit digit represents the last digit of any given number or it is obtained by getting the remainder when the given mathematical expression is divided by 10.
Unit Digit of Powers of number:
Unit Digit of powers of a power:
Example: Find the unit digit o𝐟 𝟐𝟑𝟐𝟒𝟐𝟓
Since 23 is the base number so we can treat 3 as the base
Since 3 has a cyclicity of 4, so we have to divide the power by 4 and check its nature.
Now power is 2425
Since 24 is completely divisible by 4, then 2425 will also be divisible by 4. Thus the power is of the form 4k and we know 34k = 34 = 1 (unit digit)…Ans.
Indices:
Let n be a positive integer and a be a real number, then:
an = a × a × a × a × a × a ×......................n times
where an is called "nth power of a" or "a raised to the power n"
where "a" is called the base and "n" is called index or exponent of the power an.
Laws of Indices:
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