Exercise Problem (Number System, prime number, divisible, gratest position, possible value, square, greater then, integer, odd integer, even integer, positive integer, difference)

1. If the product of 3 different positive integers is 6, then twice the sum of the integers is ?
A) 12      B) 4      C) 18      D) 36      E) None

2. The product of three different odd numbers is 15, then what is the smallest of these numbers?
A) 12      B) 4      C) 18      D) 36      E) None


3. If x is an negative odd integer & y is an positive even integer, then xy must be ?
A) Even & -ve    B) Odd & -ve      C) Even & + ve    D) Odd & + ve      E ) Can't be determined

4. The product of all the integers between -100 and 100 is ..
A) 10000      B) 99999      C) 356100      D) 0

5. If n & p are both odd numbers which of the following number must be even?
A)  N+ P     B) NP     C) np + 2      D) n + p + 1      E)  2n + p



6. If x is a positive even number, then each of the following is odd except:
A) (x + 3) (x + 5)    B) X² + 5     C) X² + 6 + 9   D) 3X² + 4    E) None

7. If positive integers x & y are not both odd, which of the following must be even?
A) xy    B) x + y    C) x-y    D) x + y - 1     E)  2(x + y ) -1

8. If a & b are both positive even integers, which of the following must be even ?
I. a^b        II. (a + b)²      III. a^{2 + 1}  
A) I only      B) I & II      C) I & III       D) I, II, III

9. If 3x represents an odd integer, which of the following represents the next larger odd integer?
A) 3(X + 1)     B) 3(x+2)      C) 3(x+ 3)      D) 3x+2      E) 3(x+2) +1

10. Which of the following is the product of two positive integers whose sum is 3 ?
A) 0      B) 1      C) 2      D) 3      E)  4

11. If the integer x is divisible by 5 but not by 2 then which one of the following is never an integer? 
A) x/24     B) x+1/2     C) X³ /3     D) x/7     E) X²/3

12. If p & q are primes greater than 2, which of the following must be true?
I. p + q is even.
II. pq is odd.
III. p² - q²  is even.
A) I only     B) II only      C) I & II      D) I & III      E) I, II & III

13. given that 
a) Square of k is odd
b) 2k is even
The integer k is :
A) odd      B) even      C) Both odd & even      D) Can't be determined

14. If -7≤ x ≤ 7  and 0 ≤ y ≤ 12, what is the greatest possible value of y - x?
A) -19      B) 5     C) 7       D) 17      E) 17

15. The number of primes divisible by 1 is ?
  A) 100      B)  200      C) 10000    D) Infinite


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Comparison Questions (important for GRE candidates)

The following 5 problems are called quantitative comparison questions. These are very important for GRE candidates. The others can practice to improve their skills. These questions consists of two questions consists of two questions; one in column A and one is column B. You are to compare to two quantities  and decide whether. 

A) The quaantity in column A is greater
B) The quantity in column B is greater
C) The two Quantities are equal
D) The relationship can't be determined from the information given

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Solve Example 4

23. If 1.4 is halfway between two points x & y what are the possible values of x & y ?
A) -1.4 AND 2.4    B) -1 & 2    C) -0.3 & 3.1    D)   0.15 AND 1.55
Ans: 1.4 is halfway between x and y means that in the number line we have a situation like

 So, 1.4-x=y-1.4 has to hold. This equation holds only for option (C) Ans: C

24. If x, w, z, correspond to four numbers -3, 1/2, -4,2 but necessarily in the same order, what is the largest possible value of the expression (wz/y)(z)2?

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Solve Examples 3

16. If m is an even integer & n is an odd integer & both are positive numbers then which of the following must be even ?

 A) m ² +n ²  B) mn + n ²  C) m ² + n ² D) mn+m ²

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Solve Examples 2

9. The difference between √150  and √54 is
A) 2√6  B) 16√6   C) 9√6   D)  6√2    E) 8√6  

Ans: √150 = √150x10 = √5x5x3x2  = 5√6
       √54 = √6x9  = 3√6  
      :. √150  - √54  = 5√6 - 3√6   = 2√6 
Ans: (A) 

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Solve Examples 1

1. Which of the following is a natural number?
A) -9    B)0    C) 1    D)   5.25
Ans: Zero, negative integers & fractions are not natural numbers. Only positive integers are natural numbers. Ans: (C)

2. Which of the following is a prime number?
A) 2   B) 0   C) 1   D) 143
Ans: 143/11=13. 0&1 are not primes by definition. 2 is a prime. Hence the answer is A.

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Number System: Prime number, Odd & Even integers, General representation of odd & even integer

Prime Number: Any integer greater than 1 which is divisible only by 1 and itself is a prime number. 2,3,5,7 are prime numbers but 9,15,27 are not primes since these are divisible by 3. remember 0 &1 are not primes.So, how do we determine whether a number is prime or not ?

One method is to divide it by all primes less than or equal to the square root of that number. If any of the divisions result in an integer, then the original number is not a prime.

For Example:  Let us try to find out if 167 is prime or not.
√167=12.92...............
So, we will attempt to divide 167 by 2,3,5,7,11. 167/2=83.5; 167/5=33.4; 167/7=23.8;167/11=15.18. Since 167 is divisible by none of 2,3,5,7,11 it is a prime.

Odd & Even integers: If an integer is divisible by 2, it is called an even integer, otherwise it is odd.
Even integers: 0,2,14,-4,72,-356,....etc.
Odd integers: 1,3,13,17,-5,-31,...etc.

General representation of odd & even integer: Any even integer can be written as '2n' where 'n' is any integer. So 4=2×2, 56=2×28, 2×3=6
Again any odd integer can be written as 2n+1, where n is any integer.
If n= 3, 2×3+1=7 (odd number)
    n=6, 2×6+1=13 (odd number)
    n=-2, 2×(-2)+1= -3 (odd number)

So, is odd + even = odd or even?
Is odd × even = odd or even?
Is even × even = odd or even?
Is odd × odd = odd or even?

3 + 2 = 5, odd + even = odd
3 × 2 = 6, odd × even = even
2 + 2 = 4, even + even = even
2 × 2 = 4, even  ×even = even
3 × 3 + 9, odd × odd = odd

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Number System: Integer, Real Number, Important properties of number line, Natural Number, Rational & Irrational number, Absolute value

Integer: Any number without a fractional value is called an integer. Numbers like 1,2,3,4,5,11,12,143,234....are positive integers. Numbers like -1,-2,-3,-4,-23,-25,-234....are negative integers. Zero (0) is an integer that is neither positive nor negative. 1.12,6.5,12.7 etc. are not integers but all the above numbers are real numbers.

Real Number: areal number is a number that  can be represented by a point on the number line.


For Example: -1>-2; 0>-1/2;0<1/2;3>-7;12>5

Natural Number:  All the positive integers are called natural number i.e. 1,2,3,4,5........... are the natural numbers. Natural numbers do not include zero, fraction or negative numbers. This set of numbers is called natural number because in the nature we find only positive integers- no fraction, on negative numbers.

Rational & Irrational numbers: A rational number is any number that can be written as a ratio of two integers. In other words, a number is rational if we can write it as a fraction where both the numerator & denominator are integers.

Example: 5,4,6,7/31,90/29

All the integers are rational numbers. Because any integer "N" may be expressed as "N/1". They have an infinite number of digits after the decimal. They are called irrational numbers.

Absolute value: The absolute value of a number 'a' is the distance between 'zero' & 'a' on the number line and is denoted by \a\.
e.g \7\=7; \-7\=7; \2.1\=2.1; /-2.1/=2.1; 

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Preface

We are very much thankful to almighty. At last this Website has come to light after a long walk.
This website is designed with a view to tutoring the examinees who are sitting for various competitive exams (BCS Preli, IBA, MBA, EMBA, NSU etc.)  in Mathematics. In these exams the basic understanding of elementary mathematics is tested. This Website is so designed that you will be able to find out what types of maths are recurrent,

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