Number Play: Thinking Puzzles

Divisibility rules work like machines, digit sums give remainders, and remainders by 7 give the day of the week.

Key ideas

A number is divisible by 3 (or 9) when the sum of its digits is.
The remainder on dividing by 3 or 9 is the same as the remainder of the digit sum.
To make the largest number divisible by 9 from given digits, pick digits whose sum is a multiple of 9, then arrange them from largest to smallest.
Days of the week repeat every 7 days, so only the remainder of n ÷ 7 matters.
The remainder of a product is the product of the remainders, divided again.
Multiples of both a and b are the multiples of their LCM.

Divisible by 2, 5, 10	last digit even; 0 or 5; 0
Divisible by 4, 8	last two digits by 4; last three digits by 8
Divisible by 6	by 2 and by 3
Remainder by 9	4,357: 4 + 3 + 5 + 7 = 19 → remainder 1
Day after n days	move on (remainder of n ÷ 7) days: 100 days from Monday is Wednesday
Multiples of a or b up to N	multiples of a + multiples of b − multiples of the LCM

Using the digit-sum rule for 4 or 8, where only the last digits matter.
Counting n days from today by adding n days to the day’s number without taking the remainder by 7.
Using a × b instead of the LCM when a and b share a factor.

Using four of the digits 8, 1, 6, 3, 9 once each, write the largest 4-digit number divisible by 3.

Working: choose four digits whose sum is a multiple of 3 and arrange them from largest to smallest

Answer: 9861