Divisibility

9 curated · 5 from open sources

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Divisibility means one integer divides another with no remainder: $a mod b$ is the remainder when $a$ is divided by $b$ . The key trick for this chapter is to work with remainders instead of full numbers. If $a mod m = r$ and $b mod m = s$ , then $(a+b) mod m = (r+s) mod m$ and $(ab) mod m = (rs) mod m$ . Powers repeat in cycles: compute $a^1, a^2, a^3, mod m$ until a remainder repeats, then use the cycle length to find any higher power. For divisibility by 9, a number has the same remainder mod 9 as the sum of its digits, because $10 mod 9 = 1$ . Squares mod 4 are always 0 or 1, since $(2 k)^{2} = 4 k^{2}$ and $(2 k + 1)^{2} = 4 k^{2} + 4 k + 1$ .

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Curated

#	Title	Difficulty	Source	Tags
13	Squares leave a traceDivisibility	Easy	Lemma	ParityModular arithmetic
24	The nine testDivisibility	Easy	Lemma	Digit sumsDivisibility by 9
36	Every square is a sum of oddsOdd numbers	Easy	Lemma	Square numbersPairing
567	A greatest common divisorDivisibility	Easy	Lemma	Divisibility
568	A least common multipleDivisibility	Easy	Lemma	Divisibility
570	A remainderDivisibility	Easy	Lemma	Divisibility
571	Odd divisorsDivisibility	Easy	Lemma	Divisibility
572	Factor 1001Divisibility	Easy	Lemma	Divisibility
1125	Divisibility by eight via last three digitsDivisibility	Easy	Hall & Knight	modular arithmeticpowers of ten

From open sources

Showing everything, including problems with no answer to check, a missing figure, or a near-copy elsewhere. Show only complete problems

#	Title	Difficulty
11356	If Mr	Easy
11485	Lucy lost one-third of her money	Easy
14937	Hamza has several empty buckets of different sizes, holding either 3, 5…	Easy
15856	Four days ago, Carlos bought a bag of gummy worms	Easy
17474	Janet’s ducks lay 16 eggs per day	Easy