Maths-

General

Easy

Question

If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is $r squared s to the power of 4 t squared$ , then the number of ordered pair (p, q) is-

224
225
252S
256

hint Hint:

At least one of the p and q, must have r,s and t in their prime factorization.
Now, we have to consider each case for selecting the number of ways for power of $r, s and t$ So, in this way we have to approach the problems.

The correct answer is: 225

Step by step solution :
It is given that $L C M space left parenthesis p comma q right parenthesis space equals space r squared s to the power of 4 t squared$
That is, at least one of $p$ and $q$ must have $r squared$ $s to the power of 4$ and $t squared$ in their prime factorizations.
Now, consider the cases for power of $r$ as follows:
Case 1 : p contains $s to the power of 4$ then q has $s to the power of K$ with K = (0,1,2,3)
That is, number of ways $=4$ .
Case 2 : q contains $s to the power of 4$ then q has $s to the power of K$ with K = (0,1,2,3)
That is, number of ways $=4$ .
Case 3 : Both p and q contains $s to the power of 4$
Then, number of ways $= 1$ .
Therefore, exponent of $r$ may be chosen in $2 + 2 + 1 = 5$ ways.
Similarly, exponent of s may be chosen in $4 + 4 + 1 = 9$ ways and
Exponent of $s$ may be chosen in $2 + 2 + 1 = 5$ ways
Thus, the total number of ways is:
$5 space cross times 9 space cross times 5 space equals space 225$
Hence, the number of the ordered pair $(p, q)$ is $225$ .

Be careful about selecting the r s and t for this particular problem. Because number ways for each selection of r, s and t there are 3 cases. Because there are two values given

p, q . Hence to calculate the total number of ways is to multiply the number of ways for r, number of ways for s and number of ways for t. So, in this way to solve this particular type of problem.

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Can’t find what you’re looking for?

If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is $r squared s to the power of 4 t squared$ , then the number of ordered pair (p, q) is-

224
225
252S
256

At least one of the p and q, must have r,s and t in their prime factorization.
Now, we have to consider each case for selecting the number of ways for power of $r, s and t$ So, in this way we have to approach the problems.

The correct answer is: 225

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Have a query? Ask a Tutor!!

Can’t find what you’re looking for?

If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is , then the number of ordered pair (p, q) is-

224 225 252S 256

At least one of the p and q, must have r,s and t in their prime factorization.Now, we have to consider each case for selecting the number of ways for power of r, s and t . So, in this way we have to approach the problems.

The correct answer is: 225

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At least one of the p and q, must have r,s and t in their prime factorization.
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