- (a + b)(a – b) = (a2 – b2)
- (a+ b)2 = (a2 + b2 + 2ab)
- (a– b)2 = (a2 + b2 – 2ab)
- (a+ b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
- (a3+ b3) = (a + b)(a2 – ab + b2)
- (a3– b3) = (a – b)(a2 + ab + b2)
- (a3+ b3 + c3 – 3abc) = (a + b + c)(a2 + b2 + c2 – ab – bc – ac)
- When a+ b + c = 0, then a3 + b3 + c3 = 3abc
Types of Numbers
Counting numbers 1,2,3,4,5,…1,2,3,4,5,… are called natural numbers
All counting numbers together with zero form the set of whole numbers.
(i) 0 is the only whole number which is not a natural number.
(ii) Every natural number is a whole number.
All natural numbers, 0 and negatives of counting numbers i.e., …,?3,?2,?1,0,1,2,3,……..,?3,?2,?1,0,1,2,3,….. together form the set of integers.
(i) Positive Integers: 1,2,3,4,…..1,2,3,4,….. is the set of all positive integers.
(ii) Negative Integers: ?1,?2,?3,…..?1,?2,?3,….. is the set of all negative integers.
(iii) Non-Positive and Non-Negative Integers: 0 is neither positive nor negative.
So, 0,1,2,3,….0,1,2,3,…. represents the set of non-negative integers,
while 0,?1,?2,?3,…..0,?1,?2,?3,….. represents the set of non-positive integers.
A number divisible by 2 is called an even number, e.g.,2,4,6,82,4,6,8, etc.
A number not divisible by 2 is called an odd number. e.g.,1,3,5,7,9,11,1,3,5,7,9,11, etc.
A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.
- Prime numbers up to 100 are :2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.:2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.
- Prime numbers Greater than 100: Let pp be a given number greater than 100. To find out whether it is prime or not, we use the following method:
Find a whole number nearly greater than the square root of pp. Let k>?jpk>?jp. Test whether pp is divisible by any prime number less than kk. If yes, then pp is not prime. Otherwise, pp is prime.
Example: We have to find whether 191 is a prime number or not. Now, 14>V19114>V191.
Prime numbers less than 14 are 2,3,5,7,11,13.2,3,5,7,11,13.
191 is not divisible by any of them. So, 191 is a prime number.
VII. Composite Numbers
Numbers greater than 1 which are not prime, are known as composite numbers, e.g., 4,6,8,9,10,12.4,6,8,9,10,12.
(i) 1 is neither prime nor composite.
(ii) 2 is the only even number which is prime.
(iii) There are 25 prime numbers between 1 and 100.
- 3. Remainder and Quotient
“The remainder is rr when pp is divided by k” means p=kq+rp=kq+r the integer qq is called the quotient.
For instance, “The remainder is 1 when 7 is divided by 3” means 7=3?2+17=3?2+1. Dividing both sides of p=kq+rp=kq+r by k gives the following alternative form pk=q+rkpk=q+rk
The remainder is 57 when a number is divided by 10,000. What is the remainder when the same number is divided by 1,000?
(A) 5 (B) 7 (C) 43 (D) 57 (E) 570
Since the remainder is 57 when the number is divided by 10,000, the number can be expressed as 10,000n+5710,000n+57, where nn is an integer.
Rewriting 10,000 as 1,000?101,000?10 yields 10,000n+57=1,000(10n)+5710,000n+57=1,000(10n)+57
Now, since nn is an integer, 10n10n is an integer. Letting 10n=q10n=q , we get
Hence, the remainder is still 57 (by the p=kq+rp=kq+r form) when the number is divided by 1,000. The answer is (D).
Method II (Alternative form):
Since the remainder is 57 when the number is divided by 10,000, the number can be expressed as 10,000n+5710,000n+57. Dividing this number by 1,000 yields
10,000n+57100010,000n+571000 =10,000n1000+571000=10,000n1000+571000 =10n+571000=10n+571000
Hence, the remainder is 57 (by the alternative form pk=q+rkpk=q+rk ), and the answer is (D).
- Even, Odd Numbers
A number n is even if the remainder is zero when nn is divided by 2:n=2z+02:n=2z+0, or n=2zn=2z.
A number nn is odd if the remainder is one when nn is divided by 2:n=2z+12:n=2z+1.
The following properties for odd and even numbers are very useful – you should memorize them:
even * evenodd * oddeven * oddeven + evenodd + oddeven + odd=even=odd=even=even=even=oddeven * even=evenodd * odd=oddeven * odd=eveneven + even=evenodd + odd=eveneven + odd=odd
If nn is a positive integer and (n+1)(n+3)(n+1)(n+3) is odd, then (n+2)(n+4)(n+2)(n+4) must be a multiple of which one of the following?
(A) 3 (B) 5 (C) 6 (D) 8 (E) 16
(n+1)(n+3)(n+1)(n+3) is odd only when both (n+1)(n+1) and (n+3)(n+3) are odd. This is possible only when nn is even.
Hence, n=2mn=2m, where mm is a positive integer. Then,
=4 * (product of two consecutive positive integers, one which must be even)=4 * (product of two consecutive positive integers, one which must be even) =4 * (an even number), and this equals a number that is at least a multiple of 8=4 * (an even number), and this equals a number that is at least a multiple of 8
Hence, the answer is (D).
Order of Magnitude
Order of Magnitude is the quantity of powers of 10 that there are in a number (or the number of powers of 0.1 in a negative number). Order of magnitude is usually written as 10 to the nth power. The n represents the order of magnitude. If you raise a number by one order of magnitude, you are basically multiplying that number by 10. If you decrease a number by one order of magnitude, you are basically multiplying that number by 0.1.
1. How many orders of magnitude are in 1,000,000 (one million)?
When you move the decimal to the left six times and stop just before the first number, you can get the order of magnitude of the large number.
Move the decimal to the left six times
So, there are six orders of magnitude of 1,000,000, meaning that you can multiply 10 six times and get 1,000,000.