Chapter 3 looked inside numbers for factors and ended with a master key, the prime fingerprint. Factors set the stage, divisibility tests gave quick eyes, primes were atoms, and GCD, LCM, and divisor counts all followed.
Key idea Factors and multiples are the same relationship stated two ways. A factor of n n n n n n d d n ÷ d n ÷ d 1 1 0 0
Key idea Closure and chaining. Add or subtract two multiples of n n n n n n closure . If a a b b b b a a b b b b a a chaining , and it works only toward the smaller factor. Every divisibility test in this chapter uses these two ideas.
With that structure in hand, the next question was practical. Must you divide a big number out the long way to know whether some small number goes into it, or can you glance at the digits and simply know? Place value gave the answer.
Key idea Divisibility tests come from place value. Split a number so that one part is a multiple of the divisor, drop that part by closure, and test the small part left behind. The last digit determines 2 2 5 5 10 10 4 4 8 8 100 = 4 × 25 100 = 4 × 25 1000 = 8 × 125 1000 = 8 × 125 9 9 3 3 11 11
Key idea Combine tests only along coprime factors. Split a composite divisor into coprime parts and test each. A split like 12 = 2 × 6 12 = 2 × 6
Testing for small factors leads to the numbers that have no smaller factors at all, the primes.
Key idea A prime has exactly two divisors. A composite has more, 1 is neither, and 2 is the only even prime. To test primality, divide by the primes up to
n n and stop there. Any factor above
n n has a smaller partner you already checked.
Key idea Every number has exactly one prime factorization (Fundamental Theorem of Arithmetic). Written with exponents like 360 = 2 3 × 3 2 × 5 360 = 2 3 × 3 2 × 5
Once one number is broken into primes, two numbers can be compared prime by prime.
Key idea Read the GCD and LCM off the two factorizations. GCD takes each shared prime to its lower power. LCM takes every prime to its higher power. They satisfy gcd × lcm = a × b g cd× lcm = a × b
Key idea The factorization gives the divisor count. For n = p a q b ⋯ n = p a q b ⋯ ( a + 1 ) ( b + 1 ) ⋯ ( a + 1 ) ( b + 1 ) ⋯
Prime factorization settled divisibility, perfect powers, GCD, LCM, and divisor counts in this chapter. Chapter 4 turns to fractions, where GCD reduces a fraction and LCM gives a common denominator.
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