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Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

The quotient when  $1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } + \dots + x ^ { 34 }$  is divided by  $1 + x + x ^ { 2 } + x ^ { 3 } + \dots + x ^ { 17 }$  is

  1. $x ^ { 17 } - x ^ { 15 } + x ^ { 13 } - x ^ { 11 } + \dots + x$
  2. $x ^ { 17 } + x ^ { 15 } + x ^ { 13 } + x ^ { 11 } + \dots + x$
  3. $x ^ { 17 } + x ^ { 16 } + x ^ { 15 } + x ^ { 14 } + \dots + 1$
  4. $x ^ { 17 } - x ^ { 16 } + x ^ { 15 } - x ^ { 14 } + \dots - 1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$1+x^2+x^4+\cdots+x^{34}=\dfrac{1-x^{36}}{1-x^2}$

$1+x+x^2+x^3+\cdots+x^{17}=\dfrac{1-x^{18}}{1-x}$

$\implies \dfrac{1+x^2+x^4+x^6+\cdots +x^{34}}{1+x+x^2+\cdots+x^{17}}=\dfrac{1-x^{36}}{1-x^{18}}\times \dfrac{1-x}{1-x^2}=\dfrac{1+x^{18}}{1+x}=x^{17}-x^{16}+x^{15}+\cdots-1$
Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

The number of positive n in the range $12\le n\le 40$ such that the product (n -1) (n -2).... 3.2.1 is not divisible by n is :

  1. 5

  2. 7

  3. 13

  4. 14
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The product (n-1)! is divisible by n if n is composite and n > 4 (Wilson's Theorem related logic). For n in [12, 40], (n-1)! is not divisible by n only if n is prime. The primes between 12 and 40 are 13, 17, 19, 23, 29, 31, 37. There are 7 such numbers.

Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

Mark the correct alternative of the following.
The product of a whole number(other than zero) and its successor is?

  1. An even number

  2. An odd number
  3. Divisible by $4$
  4. Divisible by $3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Consider whole number is $a$


Case-I

$a$ is odd
So $a=2n+1$
Successor $a\neq 1$
                  $=2n+1+1$
                  $=2n+2$
product $=(2n+1)(2n+2)$ 
              $=2(2n+1)(n+1)$
              $=2r\rightarrow Even$

Case - II
$a$ is even
So $a=2n$
Successor $=2n+1$
Product $=2n(2n+1)$
              $=2r\rightarrow\,even$

So, The product of a whole number(other than zero) and its successor is an even number.

Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

Mark the correct alternative of the following.
The product of the predecessor and successor of an odd natural number is always divisible by?

  1. $2$
  2. $4$
  3. $6$
  4. $8$
Reveal answer Fill a bubble to check yourself
A,B Correct answer
Explanation

odd Natural Number

$2n+1$

Predecessor $=2n+1-1$
                      $=2n$

Successor     $=2n+1+1$
                      $=2n+2$


Product $=2n.(2n+2)$
              $=2n.2(n+1)$
              $=4.n(n+1)$
             $=4r$
                   $\downarrow$
          divisible by $4,2$

The product of the predecessor and successor of an odd natural number is always divisible by $4,2$.