Tag: co-prime numbers

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Questions Related to co-prime numbers

Multiple choice maths be my multiple, i'll be your factor co-prime numbers lcm lowest common multiple (l.c.m.)

Solve the given exponent:
$\sqrt[4]{12} \times \sqrt[7]{6}$  

  1. $2^{\frac{9}{14}}\times 3^{\frac{11}{28}}$
  2. $3^{\frac{9}{14}}\times 2^{\frac{11}{28}}$
  3. $2^{\frac{1}{14}}\times 3^{\frac{1}{28}}$
  4. $3^{\frac{1}{14}}\times 2^{\frac{1}{28}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\sqrt[4]{12} \ \times \ \sqrt[7]{6}$


$=(12)^{\frac{1}{4}} \ \times \ (6)^{\frac{1}{7}}$

$=(2\times2\times3)^{\frac{1}{4}} \ \times \ (2\times3)^{\frac{1}{7}}$

$=(2^2\times3)^{\frac{1}{4}} \ \times \ (2\times3)^{\frac{1}{7}}$

$=2^{2\times(\frac{1}{4})}\times 3^{\frac{1}{4}} \ \times2^\frac{1}{7}\times3^\frac{1}{7}$

$=2^{\frac{1}{2}}\times 3^{\frac{1}{4}} \ \times2^\frac{1}{7}\times3^\frac{1}{7}$

$=2^{(\frac{1}{2}+\frac{1}{7})}\times 3^{(\frac{1}{4}+\frac{1}{7})}$-----If base is same, then their powers can be added, by product law.

$=2^{\frac{9}{14}}\times 3^{\frac{11}{28}}$

Option A.

Multiple choice maths be my multiple, i'll be your factor co-prime numbers lcm lowest common multiple (l.c.m.)

Choose the most appropriate option.
The traffic lights at three different signal points change after every $45$ seconds, $75$ seconds and $90$ seconds respectively. If all change simultaneously at $7:20:15$ hours, then they will change again simultaneoulsy at.

  1. $7:27:30$ hours
  2. $7:28:00$ hours
  3. $7:27:50$ hours
  4. $7:27:45$ hours
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
The $3$ signals (at $3$ points) change every $45 s,\, 75 s,\, 90 s$

So, they will change simultaneously for a common time, which is the common multiple or $L.C.M$ of the three

$\Rightarrow$  $45 = 5\times 9 = 3^2 \times 5$ 

$\Rightarrow$  $75 = 3\times 25 = 3\times 5^2$

$\Rightarrow$  $90 = 9\times 10 = 2\times 3^2\times 5$

$L.C.M= 2 \times 3^2 \times 5^2 = 2\times 9\times 25 = 450s$

So, they will change simultaneously every $450s$ or $7\,mins\,30 \,sec$

$\Rightarrow$  So, next they will change together at $7:27:45$ hours.
Multiple choice maths be my multiple, i'll be your factor co-prime numbers lcm lowest common multiple (l.c.m.)

the first four common multiple of numbers $6,8,10$ are

  1. $10,20,30,40$
  2. $120,240,360,480$
  3. $8,40,80,120$
  4. $6,60,120,240$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$6 = 2\times3$
$8 = 2^{3}$
$10 = 2\times5$

$\Rightarrow$ LCM of $6,8,10 = 2^{3}\times3\times5 = 120$

$\therefore 120$ is the least common multiple of $6,8,10$. Thus, all multiples of $120$ are common multiples of $6,8$ and $10$.

$\therefore$ First four common multiples $= 120,240,360,480$