Divisibility, primes, and digit tricks
What is the smallest positive integer that is divisible by every whole number from 1 to 10?
Find the smallest positive integer that leaves a remainder of 1 when divided by 2, 3, 4, 5, or 6, and is exactly divisible by 7.
Take any prime number greater than 3, square it, and subtract 1. Show that the result is always divisible by 24.
Pick any 2-digit number and subtract the number formed by reversing its digits (e.g., 73 − 37 = 36). The result is always divisible by what number?
What is the smallest 5-digit positive integer whose digits are all different and sum to 25?