Puzzles

Some numbers can be written as the product of two or more consecutive integers, for example:

What is the smallest three-digit number that can be written as the product of two or more consecutive integers?

The sum of three integers is 51. The product of the same three integers is 836. What is the product of largest integer and the second-largest integer?

When talking to someone about this Advent calendar, you told them that the combination of XMAS and MATHS is GREAT. They were American, so asked you if the combination of XMAS and MATH is great; you said SURE. You asked them their name; they said SAM.

Each of the letters E, X, M, A, T, H, S, R, U, and G stands for a different digit 0 to 9. The following sums are correct:

Today's number is SAM. To help you get started, the letter T represents 4.

There are six 3-digit numbers with the property that the sum of their digits is equal to the product of their digits. Today's number is the largest of these numbers.

There are 5 ways to make 30 by multiplying positive integers (including the trivial way):

Today's number is the number of ways of making 30030 by multiplying.