Find the proportion of primes p in the interval from 10^12 and 10^12 + 1000 such that p-1 is B= 105 power-smooth. Please show in a step-by-step format.

Find the x^2y^3 term in the expansion of (3x + y)^5. Simplify your answer so it does not include any binomial coefficients or factorials

Arrange the functions √n, 1000 log n, n log n, 2n!, 2n, 3", and Multiple Choice √n, 1000 log n, n log n, 2", 3¹, √n, n² 1000 log n, n log n, √n, 2n!, 2n, 3", 1,000,000 n² 1,000,000 n² 1,000,000 2n! n² 1000 log n, √n, n log n, 1,000,000 1000 log n, n log n, 2n!, 2", 3n 2", 3", 2n! n² 1,000,000 in a list so that each function is big-O of the next function.

Informal Exercise 24. Give short cuts for finding the remainder of the number [100010453000001]s when dividing by 7,8 or 9.