| Problem 0.30: Suppose x is a perfect square and its root is y. What is the result of dividing x by y? Problem 0.31: Suppose x, y and z are number siblings from the prime number family. y is the middle sibling, a prime twin of both its older sibling, z and its younger sibling, x who is the sixth largest prime number. Identify x, y and z. Problem 0.32: Let x, y be members of an Abelian Group, G. Let the group operation be denoted by the additive operator "+". (a) Indicate the neutral element in G (b) Express x in terms of itself and the neutral element (c) True or False: (i) For z = x+y, z is unique and in G (ii) (x+y)+z = x+(y+Z), for x, y, and z in G (iii) x+y = y+x, for x,y in G Problem 0. 33: x and y are prime numbers. x is also an even number. y wants to be like x. It can be x, if its value is reduced by the value of the neutral element of the Abelian group under the multiplicative operation. What is the value of y ? Problem 0.34: x and m are integers, m>0. What is the value of m, if the third least Mersenne prime is equal to max(x modulo m). Suppose y is a perfect cube. What is the restriction on y, if m is to contain y? Problem 0.35: x is an integer. S = { 12, 17, 22, 27, 32 } is a subset of (x mod 5), the congruence class of intergers modulo 5 congruent to x modulo 5. What is the minimum value of x, the representative of the congruence class (x mod 5)? Problem 0.36: G and G' are finite groups under multiplication. The elements of G are the first 4 Merssene primes. The elements of G' are the first 4 positive integers. Under what scenarios are G and G' isomorphic? Problem 0.37: G and G' are finite groups under multiplication. The elements of G are the first 4 Merssene primes. The elements of G' are the first 4 positive integers. What are the generators of the subgroup that is the intersection of G and G' ? |
| Problem 0.38: P, Q are non-zero polynomials in one indeterminate X and over the ground field R (i.e. the coefficients of P and Q are real numbers).
What is deg(PQ)?
Problem 0.39: P is a polynomial of degree 0. If the function of x, f(x) = P, what is the gradient of the graph of f ? Problem 0.40: P = x2 + 1, is irreducable over the ground field of real numbers. Is P irreducable over the ground field of complex numbers? Problem 0.41: P = (x-a)Q is a polynomial. Q is a polynomial of degree m. What is the maximum number of distinct roots associated with P. Problems by Peter O. Sagay. |
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