Theorem 1.3 : Let p be a prime number. If p divides a^2 , then p divides a, where a is a positive integer.

The contrapositive of the statement, 'If x is a prime number and x divides ab then x divides a or x divides b", can be symnolically represented using logical connectives, on appropriately defined statements p, q, r, s as

Let P = { x: x is a positive integer , x lt 6 }

Let p, q be prime numbers such that non n^(3pq)-n is a multiple of 3pq for all positive integers n. Find the least possible value of p + q.

Three different numbers are selected at random from the set A={1,2,3,……..,10} . Then the probability that the product of two numbers equal to the third number is (p)/(q) , where p and q are relatively prime positive integers then the value of (p+q) is :

If a is defined by f (x)=a_(0)x^(n)+a_(1)x^(n-2)+a_(2)x^(n-2)+...+a_(n-1)x+a_(n) where n is a non negative integer and a_(0),a_(1),a_(2),…….,a_(n) are real numbers and a_(0) ne 0, then f is called a polynomial function of degree n. For polynomials we can define the following theorem (i) Remainder theorem: Let p(x) be any polynomial of degree greater than or equal to one and 'a' be a real number. if p(x) is divided by (x-a), then the remainder is equal to p(a). (ii) Factor theorem : Let p(x) be a polynomial of degree greater than or equal to 1 and 'a' be a real number such that p(a) = 0, then (x-a) is a factor of p(x). Conversely, if (x-a) is a factor of p(x). then p(a)=0. THe polynomials P(x) =kx^(3)+3x^(2)-3 and Q(x)=2x^(3) -5x+k, when divided by (x-4) leave the same remainder. Then the value of k is

If a is defined by f (x)=a_(0)x^(n)+a_(1)x^(n-2)+a_(2)x^(n-2)+...+a_(n-1)x+a_(n) where n is a non negative integer and a_(0),a_(1),a_(2),…….,a_(n) are real numbers and a_(0) ne 0, then f is called a polynomial function of degree n. For polynomials we can define the following theorem (i) Remainder theorem: Let p(x) be any polynomial of degree greater than or equal to one and 'a' be a real number. if p(x) is divided by (x-a), then the remainder is equal to p(a). (ii) Factor theorem : Let p(x) be a polynomial of degree greater than or equal to 1 and 'a' be a real number such that p(a) = 0, then (x-a) is a factor of p(x). Conversely, if (x-a) is a factor of p(x). then p(a)=0. The remainder when the polynomial P(x) =x^(4)-3x^(2) +2x+1 is divided by x-1 is

Seven digits from the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are written in random order. Let the probability that this number is divisible by 9 be p=a/b . Where a, b are relatively prime positive integers, then the value of a + b is ……….

Prove that a positive integer n is prime number, if no prime p less than or equal to sqrtn divides n.

Let P(n) be the statement "7 divides 2^(3n)-1 ." What is P(n+1) ?

Let the sum sum_(n=1)^(9)1/(n(n+1)(n+2)) written in the rational form be p/q (where p and q are co-prime), then the value of [(q-p)/10] is (where [.] is the greatest integer function)