Question

P, Q and R are three natural numbers such that P and Q are primes and Q exactly divides PR .Then out of the following the correct statement is \[\]
A.Q exactly divides R \[\]
B. P exactly divides R\[\]
C. Q exactly divides QR\[\]
D. P exactly divides PQ \[\]

Answer 1

Hint: We recall the definitions of dividend, divisor, quotient, and remainder. We recall that a divisor is said to exactly divide the dividend only when the remainder after division is zero. We use the property that if a prime number divides the product of two numbers then it has to divide at least of the two numbers. \[\]


Complete step by step answer:
We know that in the arithmetic operation of division the number we are going to divide is called the dividend, the number by which divides the dividend is called the divisor. We get a quotient which is a number of times the divisor is of dividend and also remainder obtained at the end of the division. If the number is $ n $ , the divisor is $ d $ , the quotient is $ q $ and the remainder is $ r $ , then by Euclid’s lemma we have
\[n=dq+r\]
Here the divisor $ d $ cannot be zero. If the remainder $ r=0 $ ,we say $ d $ exactly divides the number $ n $ or $ d $ is factor $ n $ .
We call a number prime when the number has only two factors:1 and the number itself. We know that when a prime $ p $ exactly divides the product of two numbers $ m $ and $ n $ if and only if $ p $ divides at least one of the numbers between $ m $ and $ n $ .
We also know a prime cannot exactly divide another prime.
We give in the question that Q and R are three natural numbers such that P and Q are primes and Q exactly divides PR . Let us look at the options. \[\]
We see that Q is prime that exactly divides the product of two numbers P and R. Since P is also prime Q cannot exactly divide it and Q has to exactly divide R. So option A is correct.
Since it is not necessary that P has to exactly divide R or QR, options B and C are not correct.\[\]
Since a product of two numbers is always exactly divisible by the two numbers, P will exactly divide its product with Q that is PQ. So option D is correct. \[\]

Note:
 We note that we call $ n $ a multiple of $ d $ , if $ d $ is a factor $ n $ . If two numbers are exactly divided by the divisor $ d $ , then $ d $ is called a common divisor of the two numbers. Two primes cannot divide each other because they do not have any common divisor. We can also extend that if the prime $ p $ divides one of the numbers among $ {{N}_{1}},{{N}_{2}},...,{{N}_{n}} $ then it divides their product $ {{N}_{1}}\times {{N}_{2}}\times ...\times {{N}_{n}} $ .