### Question

#### If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is r^{2}t^{4}s^{2}, then the number of ordered pair (p, q) is –

- 224
- 225
- 252
- 256

### Hint:

Natural numbers known as prime numbers can only be divided by one (1) and by the number itself. In other terms, prime numbers are positive integers greater than one that only have the number itself and the number's first digit as factors. Here we have given r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is r2t4s2, then what is the number of ordered pair (p, q).

#### The correct answer is: 225

#### An ordered pair is made up of the ordinate and the abscissa of the x coordinate, with two values given in parenthesis in a certain sequence.

Now we have given the LCM as: r^{2}t^{4}s^{2}

Consider following cases:

Case 1: if p contains r^{2} then q will have r^{k}, for the value k=0,1.

So 2 ways.

Case 2: if q contains r^{2} then p will have r^{k}, for the value k=0,1.

So 2 ways.

Case 3:Both p and q contain r^{2}

So 1way.

So after this we can say that:

exponent of r=2+2+1 = 5 ways.

Similarly

exponent of t=4+4+1=9 ways.

exponent of s=2+2+1 = 5 ways.

So total ways will be:

5 x 5 x 9 = 225 ways.

Finding the smallest common multiple between any two or more numbers is done using the least common multiple (LCM) approach. A number that is a multiple of two or more other numbers is said to be a common multiple. Here we understood the concept of LCM and the pairs, so the total pairs can be 225.

## Book A Free Demo

Grade*

### Related Questions to study

#### A rectangle has sides of (2m – 1) & (2n – 1) units as shown in the figure composed of squares having edge length one unit then no. of rectangles which have odd unit length

There are 2n horizontal lines and 1 2 m vertical lines (numbered 1,2.......2n).

Two horizontal lines, one with an even number and one with an odd number, as well as two vertical lines must be chosen in order to create the necessary rectangle.

Then the number of rectangles will be:

(1+3+5+......+(2m−1))(1+3+5+......+(2n−1))=m

^{2}n

^{2}

So it is m

^{2}n

^{2}.

#### A rectangle has sides of (2m – 1) & (2n – 1) units as shown in the figure composed of squares having edge length one unit then no. of rectangles which have odd unit length

There are 2n horizontal lines and 1 2 m vertical lines (numbered 1,2.......2n).

Two horizontal lines, one with an even number and one with an odd number, as well as two vertical lines must be chosen in order to create the necessary rectangle.

Then the number of rectangles will be:

(1+3+5+......+(2m−1))(1+3+5+......+(2n−1))=m

^{2}n

^{2}

So it is m

^{2}n

^{2}.

^{n}C_{r} + ^{2n}C_{r+1} + ^{n}C^{r+2} is equal to (2 r n)

^{n}C_{r} + ^{2n}C_{r+1} + ^{n}C^{r+2} is equal to (2 r n)

The coefficient of in is

The coefficient of in is

#### How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which not two S are adjacent ?

#### How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which not two S are adjacent ?

#### The value of ^{50}C_{4} + is -

#### The value of ^{50}C_{4} + is -

#### The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is-

#### The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is-

#### A mass of strikes the wall with speed at an angle as shown in figure and it rebounds with the same speed. If the contact time is , what is the force applied on the mass by the wall

Initial momentum

Final momentum

Substituting

Force on the ball

Negative sign indicates direction of the force

#### A mass of strikes the wall with speed at an angle as shown in figure and it rebounds with the same speed. If the contact time is , what is the force applied on the mass by the wall

Initial momentum

Final momentum

Substituting

Force on the ball

Negative sign indicates direction of the force

#### If the locus of the mid points of the chords of the ellipse , drawn parallel to is then

#### If the locus of the mid points of the chords of the ellipse , drawn parallel to is then

#### If ^{n}C_{r} denotes the number of combinations of n things taken r at a time, then the expression ^{n}C_{r+1} + ^{n}C_{r –1} + 2 × ^{n}C_{r} equals-

#### If ^{n}C_{r} denotes the number of combinations of n things taken r at a time, then the expression ^{n}C_{r+1} + ^{n}C_{r –1} + 2 × ^{n}C_{r} equals-

#### The number of ways is which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question is -

#### The number of ways is which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question is -

#### An intense stream of water of cross-sectional area strikes a wall at an angle with the normal to the wall and returns back elastically. If the density of water is and its velocity is ,then the force exerted in the wall will be

Now making components of momentum along - axes and -axes. Change in momentum of water per second

By definition of force, force exerted on the Wall

#### An intense stream of water of cross-sectional area strikes a wall at an angle with the normal to the wall and returns back elastically. If the density of water is and its velocity is ,then the force exerted in the wall will be

Now making components of momentum along - axes and -axes. Change in momentum of water per second

By definition of force, force exerted on the Wall

#### The force required to stretch a spring varies with the distance as shown in the figure. If the experiment is performed with above spring of half length, the line will

Slope of force displacement graph gives the spring constant of spring

If becomes double then slope of the graph increases graph shifts towards force- axis

#### The force required to stretch a spring varies with the distance as shown in the figure. If the experiment is performed with above spring of half length, the line will

Slope of force displacement graph gives the spring constant of spring

If becomes double then slope of the graph increases graph shifts towards force- axis

#### Two small particles of equal masses start moving in opposite directions from a point in a horizontal circular orbit. Their tangential velocities are and , respectively, as shown in the figure. Between collisions, the particles move with constant speeds. After making how many elastic collisions, other than that at , these two particles will again reach the point

As the ratio of velocities of and particles are , therefore ratio of their distance covered will be in the ratio of . It means they collide at point B

After first collision at B, velocities of particles get interchanged, ., will move with and particle with

Second collision will take place at point C. Again at this point velocities get interchanged and third collision take place at point A

So, after two collision these two particles will again reach the point A

#### Two small particles of equal masses start moving in opposite directions from a point in a horizontal circular orbit. Their tangential velocities are and , respectively, as shown in the figure. Between collisions, the particles move with constant speeds. After making how many elastic collisions, other than that at , these two particles will again reach the point

As the ratio of velocities of and particles are , therefore ratio of their distance covered will be in the ratio of . It means they collide at point B

After first collision at B, velocities of particles get interchanged, ., will move with and particle with

Second collision will take place at point C. Again at this point velocities get interchanged and third collision take place at point A

So, after two collision these two particles will again reach the point A