Explore patterns in remainders and discover clever number tricks that reveal the hidden logic behind arithmetic.
When you divide numbers by a fixed divisor, the remainders often follow a repeating cycle. Spotting this cycle lets you predict remainders without doing every division.
Find the remainder when 1, 2, 3, 4, 5, 6, 7 are each divided by 3.
1, 2, 0, 1, 2, 0, 1 — the remainders repeat in a cycle of 0, 1, 2
Using the pattern, what is the remainder when 100 is divided by 3?
100 = 3×33 + 1, so the remainder is 1
Many 'magic' number tricks work because of consistent algebraic patterns. Working through the steps with a variable reveals why the trick always produces the same result.
Think of a number, double it, add 6, then divide by 2, then subtract the original number. What do you always get?
3 — because (2x+6)/2 − x = (x+3) − x = 3, no matter what x is
Think of a number, add 5, multiply by 2, subtract 10, then divide by 2. What do you get?
The original number — because (2(x+5)−10)/2 = (2x+10−10)/2 = 2x/2 = x
1. Find the remainder when 1 through 6 are divided by 4, and describe the pattern.
2. Using the pattern for dividing by 4, what is the remainder when 50 is divided by 4?
3. Think of a number, triple it, add 9, then divide by 3, then subtract the original number. What do you always get?
4. Think of a number, add 4, double it, subtract 8. What relationship does the result have to the original number?
5. Find the remainder when 1 through 5 are divided by 5, and describe the pattern.