What is the largest three-digit number with the property that the number is equalto the sum of its hundreds digit, the square of its tens digit and the cube of its unitsdigit?27.Igor wants to make a secret code of five-letter words. To make them easy to say, hefollows these two rules:(i) no more than two consonants or two vowels in succession(ii) no word to start or end with two consonantsHe rejects the letter ‘Q’ as too hard, so he has 20 consonants and 5 vowels to choosefrom. IfNis the number of code words possible, what are the first three digits ofN?28.Consider the sequencea1,a2,a3,a4,. . .such thata1= 2 and for every positiveintegern,an+1=an+pn,wherepnis the largest prime factor ofan.The first few terms of the sequence are 2,4,6,9,12,15,20. What is the largest valueofnsuch thatanis a four-digit number?29.Alattice pointin the plane is a point whose coordinates are both integers. Considera triangle whose vertices are lattice points (0,0), (a,0), and (0, b), wherea≥b >0.Suppose that the triangle contains exactly 74 lattice points in its interior, notincluding those lattice points on the sides of the triangle. Determine the sum of theareas of all such triangles.30.A polynomialp(x) is calledself-centeredif it has integer coefficients andp(100) = 100.Ifp(x) is a self-centred polynomial, what is the maximum number of integer solutionskto the equationp(k) =k3?