Divisibility

4 curated · 4 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
3	A sum that's never wholeDivisibility	Deep	Lemma	2-adic valuationParity+1
37	Zeros at the end of 100!Factorials	Deep	Lemma	Factors of 5Legendre's formula
62	The last two digits of a giantPatterns	Deep	Lemma	Cycles mod 100Powers
1132	Repeating‑digit divisorDivisibility	Deep	Hall & Knight	repeating blocksmodular arithmetic

From open sources

Complete problems only, with worked solutions first. Show everything

#	Title	Difficulty
10817	Matt needs to buy new plates for his home	Deep
11804	There are 90 students who have lunch during period 5	Deep
13325	Monica and Sheila are twins	Deep
15020	Sienna gave Bailey half of her suckers	Deep