Concise Introduction To Pure Mathematics Solutions Manual -

Digits 0–9, evens = 0,2,4,6,8, odds = 1,3,5,7,9.

But must exclude numbers starting with 0? If first digit is 0, it’s not a 4‑digit number. Count invalid: Fix first digit=0 and it’s one of the two even positions. Choose other even position (3 ways), fill that even (5 ways). Fill two odd positions (5^2). So invalid = (3\times 5\times 25 = 375). Valid = (3750 - 375 = 3375). Concise Introduction To Pure Mathematics Solutions Manual

Find remainder when (x^100) is divided by (x^2-1). Digits 0–9, evens = 0,2,4,6,8, odds = 1,3,5,7,9

Solve (3x \equiv 5 \pmod11).

Case 1: first digit odd (4 choices: 1,3,5,7,9? Actually 5 odds, but careful: first digit ≠0, so even allowed but handled separately). Better systematic: Choose positions for the two even digits: (\binom42=6) ways. Count invalid: Fix first digit=0 and it’s one

[ A\cup B = 1,2,3,4,\quad A\cap B = 2,3 ] [ A\setminus B = 1,\quad B\setminus A = 4 ] Remark : Set difference removes elements of the second set from the first.