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




      Ans:  Lets try its easy way, Let, m= 2, n=3
     (A)   m ² + n ²  = 2² +3² =  13
     (B)  mn +n ²  =    2x3 +3² =15
     (C) m ³ + n ² = 2³ + 3²  = 17
     (D) mn+m ²  = 2x3+2² =10
 The Answer is D

17. Sum of 3 different positive integers is the same as their product. Find the smallest of these 3 integers. 
A) 3   B) 2    C) 1    D) 0    E)  None
Ans: 1+2+3=6,  1x2x3=6
So, 1,2,3 are the 3 different positive integers we are looking for. Among them, 1 is the smallest. So, C is the answer.

18. If the product of 2 integers is an even number & the sum of the same two integers is an odd number, Which of the following must be true?
A) Both the integers are odd.
B) Both the integers are even.
C) One is odd & another is even.
D) One of the integers is 1.

Ans: From the properties of odd & even numbers, we know
Even + odd= odd
Even x odd= even
Shortcut: 2x2 =4,2x3=6, 2+2=4, 2+3=5
Hence the correct answer is C

19. Which of the following must be odd? 
I. The product of two even numbers
II. The product of two odd numbers
III. The sum of an odd and an even number
A) None    B) I only   C) II & III only D) I & III only    E)  I,II & III
Ans: For 2 & 4, 2x4= 8 = I wrong
For 3 & 5 , 3x5=15 = II right
 For 2&3, 2+3= 5 = III right
Ans: (C)

20. If a & b are prime numbers such that a>b, which of the following can't be true?
A) a+b is positive   B) ab is odd   C) a(a+b) is odd   D)  a-b is prime   E) a ^b   is even

Ans: A is true as all primes are greater than 1.
B can be true.
2 & 3 are primes: 2x3 = 6 which is even :. B is not true.
2 & 5 are primes: 3x5 = 15 which is odd :. B is true.
C is true for a similar reason. Let, a=3 , b=2; a(a+b) = 3(3+2) =15 (odd)
D can be true if a = 5 & b = 2. Because a-b = 3 (Prime)
E can't be true since (odd)x is always odd,
[ 3² =9, 3³ =27, 3⁴ =81, 5² =25, 5³ =125 ]
Ans: (E)

21. If 2 <x<4 and 3< y <7, What is the largest integer value of x+y?
A) 7      B) 8      C) 9      D) 10      E) 11
Ans: If x & y are integers, the largest possible value is 3+6=9 . However, note that although x + y has to be an  integer, x & y does not need to be integers.
If x = 3.8 & y = 6.2 then x+y = 10, So, the answer is D.

22. If the 1^st number in a series of consecutive odd numbers is 8 less than the last number in the series, how many numbers are there in the series ?
A) 4     B) 5    C) 7     D) 8    E) None
Ans: Let, x be the 1^st  odd number. So, the consecutive numbers are
x, x+2, x+4, x+6, x+8.
:.  There are 5 numbers in the series.   
  Ans: (B)

Posted in: Number System