Solutions Integers

 Integer Review:    Fundamentals of Integers , Rules for Multiplication , Rules for divisibility,  Division Terminology,  Even & Odd Integer,  Rules about even & odd integer, Prime Number,  Prime Factorization ,  Distinct Prime Number,  LCM & HCF,  Composite Numbers ,  Consecutive Integers

Class Questions :   Exercise 1         Exercise 2         Exercise 3        Exercise 4      Exercise 5

Practice Questions :   Solutions  

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Q.11        The quotient of   m/7   is an odd integer and there is no remainder. Which must be the correct statement from below?

                         

    A .   ‘m’     is  a  prime integer

             B.     ‘m’    is  a  consecutive integer

             C.      ‘m’    is  an  even  integer

             D.      ‘m’    is  an  odd  integer

             E.      ‘m’     is multiple of 3

Solution

      The odd integers are 1, 3, 5, 7………………….     Here it is given that the quotient of    m/7   is an odd integer that means the quotient will be 1, 3, 5, 7 ……………….

We  can refer Division Terminology in the Review Section of Integer

           

             If          m/7  = 1;           then               m = 7

                          m/7 = 3;            then            m = 21

                          m/7  = 5;             then           m= 35

That means for any value of    m/7   to be odd the value of   ‘m’ must be odd. So the correct answer choice is ‘D’.

 Q.12  What are the multiples of 3, 6, & 9  all together? Choose all the options from the following.

    A. 12

    B. 15

     C. 18

     D. 45

     E. 54

   

Solution;

If we check option choice one by one we can easily get the answer.

Multiples  of 3 =   3, 6, 9, 12, 15, 18…………………

Multiples  of   6  =   6, 12, 18, 24……………….

Multiples  of   9  =  9, 18, 27 ,36, 45,   54………………

12 is a multiple of 3 and 6 but not of 9

15  is a multiple of 3 but not of 6 and 9

18  is a multiple of  3, 6 and 9

45  is multiple of 3 and 9 but not 6

54   is a multiple of 3, 6 and 9

                           So, correct answer will be C and E.

Q.13                                  m is an even integer and n is an odd integer

                                 Quantity A                                           Quantity B

                                       3                                                                   m/n

             A.  If Quantity A is greater

             B.  If Quantity B is greater

             C. If Quantity A and Quantity B are equal

             D. The relationship cannot be determined from given information

 

 Solution;

Let us take values for m and n

If m = 2, n = 1 then m/n = 2/1 = 2 Quantity A is greater

If m = 12, n =3 then m/n = 12/3 = 4 Quantity B is greater

So, Both the Quantity A & Quantity B cannot be compared from given data, Hence

Answer is ‘D’.

 Q.14                                          Dividend ‘x’ = 36. ‘x’ is square of its quotient

                                             Quantity A                                                    Quantity B

                                           Divisor of ‘x’                                                                   6

 

       A.  If Quantity A is greater

       B.  If Quantity B is greater

       C. If Quantity A and Quantity B are equal

       D. The relationship cannot be determined from given information

 

Solution;

Here it is given that Dividend = x = 36, which is square of its quotient. That means 36 is square of its quotient which should be ‘6’. So if we divide 36 with 6 then only ‘x’ can be square of quotient.

So in above condition;

Dividend = x = 36

Divisor = 6

Quotient = 6

The value for Quantity A will be 6. Now if we compare The value of Quantity B which is also 6 , it comes out  equal so correct answer choice will be ‘C’.

                                             

Q.15

The integer m is even and integer n is even and prime. The square of ‘m’ is addition of m and n. Find the least possible value for ‘m’?

Solution;

The possible values for ‘m’ could be 2, 4, 6, 8………………………..

The value for ‘n’ will be = 2, as 2 is only even and prime number.

Now if we take ‘m’ = 2, ‘n’ = 2, then  m²  =  m x m  = 2 x 2 = 4

Now addition of       m + n = 2 + 2 = 4.

As the least possible value is being asked, so this condition justifies the asked question.

So the least possible value for ‘m’ will be =2.

                                                           

Q.16     If ‘x’ is double to the next number of sum of first three prime numbers.

                        Quantity A                                                       Quantity B

                               x                                                                          23

 

           A.  If Quantity A is greater

          B.  If Quantity B is greater

          C. If Quantity A and Quantity B are equal

          D. The relationship cannot be determined from given information

 

Solution;

The first three prime numbers are 2, 3 and 5

The Sum = 2 + 3 + 5 = 10

Next number to 10 is 11.

Double of 11, so ‘x’ = 2 x 11 = 22

If we compare Quantity A = x = 22 and Quantity B = 23, Quantity B is always greater.

The correct answer choice is ‘B’.

Q.17 If ‘x’ is even and prime.  It is one of the factors and digit at ten’s place of the number of another factor with a digit at one’s place is square of 3. What is that number?

Solution;

‘x’ is even and prime i.e. x = 2 , Now there is one integer with two factors .

One of the factors is 2. And for second factor it is given that digit at tens place is 2 and one’s place is square of 3 which is 9. So, the second factor is 29.

Hence the number = 2 x 29 = 58.

The correct answer is 58.

Q. 18          If ‘m’ is a positive integer and ‘n’ is a negative integer then which of the   following value must be true for m – n?

      A. Positive

      B. Negative

      C. Positive and even

       D. Prime

        E. Only even

Solution;

Let us take the positive value for (i) m = 1 and negative value for n = -1;

Then m-n = 1 – (-1) = 1 + 1 = 2

(ii) Now take next value for m = 3 and n = -5

Then m-n = 3 – (-5) = 3 + 5 = 8

(iii) For m = 5 and n = -4

Then m – n = 5 – (- 4) = 5 + 4 = 9

(iv)Take one more value for m = 6 and n= -8

Then m – n = 6 – (- 8) = 6 + 8 =14

Every time the result is positive integer only.

So correct answer is ‘A’.

Q. 19        If ‘m’ is multiplied to ‘n’ the result is 1. If ‘m’ is divided by ‘n’ the result is 4  what  is m + n?

A. 1/5

B.  3/4

C.  5/6

D. 1/2

E.  5/2

Solution;

                  m x n = 1 …………………………….Equation 1

                m/n   =   4   so,   m = 4n ……………………. Equation 2

         Now put value of ‘m’ in equation 1

4n x n = 1

n²  =  1/4   ;

Taking square root both side;

             

Q.20                                                                         If a is square of integer t

                                             Quantity A                                                       Quantity B

                                                        a/t                                                                      t/a

       A.  If Quantity A is greater

       B.  If Quantity B is greater

       C. If Quantity A and Quantity B are equal

       D. The relationship cannot be determined from given information

Solution;

The best rule to solve this problem is taking one positive and one negative value for given variables and compare the Quantity A with Quantity B.

Let us start by putting the values for integer‘t’.

       (i)    If t = 2 then a = 2² =  2 x 2 = 4

                      a/t   =  4/2   = 2                                         t/a    =     2/4    =  0.5

Quantity A is greater, Eliminate Choice ‘C’

       (ii)    If t = -3 then a = (-3)²  =  9

                      a/t   =  9/-3    =   – 3                                    t/a     =    -3/9    =    – 0.33

Quantity B is greater

As it is difficult to reach to any conclusion after comparing Quantity A and Quantity B, The correct answer choice will be ‘D’.