Tag: ways to multiply and divide

Questions Related to ways to multiply and divide

Given $n =1 + x$ and x is the product of four consecutive integers. Then which of the following is true?

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

  1. n is an odd numbers

  2. n is prime

  3. sometimes a perfect square

  4. All the given

Show answer
Correct Option: A,C
Explanation:

$ Let\quad x=a(a+1)(a+2)(a+3)\ Now\quad whether\quad a\quad is\quad even\quad or\quad odd,\quad the\quad four\quad consecutive\quad number's\quad product\quad \ contain\quad 2,\quad 3,\quad 4\quad as\quad factors.\ \therefore \quad x\quad is\quad divisible\quad by\quad 2\times 3\times 4=24----(1)\ Option\quad A\longrightarrow Four\quad consecutive\quad numbers\quad contain\quad two\quad even\quad numbers.\ Therefore\quad product\quad is\quad even.\therefore \quad x+1\quad should\quad be\quad odd.\ Option\quad A\quad is\quad correct.\ Option\quad B\longrightarrow x\quad is\quad divisible\quad by\quad 24\quad (from\quad 1)\ Now\quad there\quad exists\quad no\quad prime\quad of\quad the\quad form\quad 24p+1\quad where\quad p\quad is\quad a\quad natural\quad number.\ e.g.\quad \quad 29=24\times 1+5\ \quad \quad \quad \quad \quad 31=24\times 1+7\ \quad \quad \quad \quad \quad 37=24\times 1+13\ With\quad increasing\quad count\quad the\quad second\quad term\quad increases\quad because\quad difference\quad between\ primes\quad increase\quad with\quad higher\quad count.\quad Therefore\quad x+1=n\quad is\quad not\quad a\quad prime\quad under\quad the\quad \ given\quad condition.\quad For\quad a\quad expanded\quad number\quad to\quad be\quad square,\quad the\quad first\quad and\quad last\quad term\ should\quad be\quad square\quad number.   Option\quad C\longrightarrow x=a(a+1)(a+2)(a+3)\ when\quad a\quad is\quad even\quad a=2p\ \therefore \quad x=2p(2a+1)(2a+2)(2a+3)\ \quad \quad \quad =4p(2p+1)(p+1)(2p+3)\ The\quad product\quad is\quad 4\times 1\times 1\times 3=12\ If\quad we\quad add\quad 1\quad then\quad last\quad term\quad of\quad the\quad product\quad is\quad 13\ which\quad is\quad not\quad a\quad square\quad number.\ So\quad x+1=n\quad is\quad not\quad a\quad square\quad number\quad when\quad a\quad is\quad even.\ (2)\quad When\quad a\quad is\quad odd-\ x=(2p+1)(2p+2)(2p+3)(2p+4)\ The\quad last\quad term\quad of\quad the\quad product\quad is\quad 1\times 2\times 3\times 4=24.\ 24+1=25\quad is\quad a\quad square\quad term.\ \therefore \quad n=x+1\quad is\quad a\quad square\quad number\quad when\quad a\quad is\quad odd.\ \therefore \quad Option\quad C\quad is\quad correct\quad when\quad a\quad is\quad odd.\ Option\quad D\longrightarrow \quad Obviously\quad option\quad D\quad is\quad not\quad correct. $

Kunal has only $25$ paise and $50$ paise coins with him. The total amount in $50$ paise denomination is $Rs. 4$ more that the total amount in $25$ paise denomination. The number of $25$ paise coins is $20$ more than the number of $50$ paise coins. What is the total amount with Kunal?

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

  1. $Rs. 32$

  2. $Rs. 36$

  3. $Rs. 40$

  4. $Rs. 24$

Show answer
Correct Option: A

The sum of all natural members which multiples of $7$ or $3$ or both and lie between $200$ and $500$ is

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

  1. $45049$

  2. $40149$

  3. $45149$

  4. $45249$

Show answer
Correct Option: A

A number when divided by $14$ leaves a remainder of $8$, but when the same number is divided by $7$, it will leave the remainder ?

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

  1. 3

  2. 2

  3. 1

  4. can't be determined

Show answer
Correct Option: C
Explanation:

We have,

When the number is divided by $14$ it gives a remainder of $8$,

The number $= 14N + 8 (14N$ is divisible by $14)$

When same number is divided by $7$ it will give remainder $1.$

hence, this is the answer.

The sum of all two digit numbers which when divided by 4 , yield unity as remainder is 

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

  1. $1012$

  2. $1201$

  3. $1212$

  4. $1210$

Show answer
Correct Option: D
Explanation:

number should be of the form $4k+1$


Smallest 2 digit number that gives remainder $1$ when divided by $4$ $\Rightarrow$ $13 (when \ k=3)$ first term of A.P

Largest 2 digit number that gives remainder $1$ when divided by $4$ $\Rightarrow$ $97 (when \ k=24)$ last term of AP

Series: $13,17,21,....97$

$97=a+(n-1)d$

$97=13+(n-1)4$

$89=(n-1)4$

$(n-1)=21$

$n=22$

Sum of series $=\cfrac{n}{2}$[first term  + last term]

$=\cfrac{22}{2}[13+97]$

$=11\times (110)$

$=1210$

A rectangular cortyard $3.78$ metres long and $5.25$ metres wide is to be paved exactly with square tiles, all of the same size. Then the largest size of the tile which could be used for the purpose is $(n\times 3)\ cm$, then $n$ is equal to

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

  1. $5$

  2. $7$

  3. $2$

  4. $13$

Show answer
Correct Option: B

$\dfrac{20}{100}\times 1,70000=20\times 1700=34,000$

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\begin{matrix} \dfrac { { 20 } }{ { 100 } } \times 170000=20\times 1700 \ =34000\, \, \, Ans. \  \end{matrix}$

A train running at the speed of 60$\mathrm { km } / \mathrm { hr }$ crosses a pole in 9 seconds. What is the length of the train

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

  1. $150$ metres

  2. $180$ metres

  3. $324$ metres

  4. Cannot be determined

  5. None of these

Show answer
Correct Option: A
Explanation:

$Speed\>of\>train\>=\>60\>(\frac{Km}{hr})\\=(\frac{60\cdot\>1000\>m}{3600\>sec})\\=(\frac{50}{3})m/sec\\length\>crossed\>in\>9\>seconds\>=\>(\frac{50}{3})\cdot\>9\>=\>150\>m\\\therefore\>length\>of\>train\>=\>150m$

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

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

  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. $x ^ { 17 } - x ^ { 16 } + x ^ { 15 } - x ^ { 14 } + \dots - 1$

Show answer
Correct Option: D
Explanation:
$1+x^2+x^4+\cdots+x^{34}=\dfrac{1-x^{36}}{1-x^2}$

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

$\implies \dfrac{1+x^2+x^4+x^6+\cdots +x^{34}}{1+x+x^2+\cdots+x^{17}}=\dfrac{1-x^{36}}{1-x^{18}}\times \dfrac{1-x}{1-x^2}=\dfrac{1+x^{18}}{1+x}=x^{17}-x^{16}+x^{15}+\cdots-1$

What should be multiplied by $-23$ to get $575$?

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

  1. $15$

  2. $25$

  3. $-25$

  4. $35$

Show answer
Correct Option: C
Explanation:

Let $x $ be multiplied by $-23$ to get $575$


$x (-23) = 575$


$x = \dfrac{575}{-23}$

$\therefore x = -25$