Properties of Integers — Even, Odd, Prime, Divisibility

GRE Quantitative · Axiom Academy

Every integer has a hidden prime "DNA" — once you can see it, even/odd, divisibility, and prime all become the same idea. 1. The Factor Tree — a Number's Prime DNA Take 24 and start splitting it into two factors, then split each piece again, until nothing splits any further. Every branch that can't be split — a prime — becomes a leaf. Multiply all the leaves back together and you rebuild the original number. The leaves, multiplied back together 2. Even and Odd — Does a 2 Show Up in the Tree? A number is even exactly when its factor tree contains a leaf of 2 — equivalently, it's divisible by 2. A number is odd when no 2 ever appears. That's the whole distinction; everything else is a shortcut for checking it fast. Divisible by 2. Quick test: ends in 0, 2, 4, 6, or 8. Not divisible by 2. Quick test: ends in 1, 3, 5, 7, or 9. Every even number's tree splits off a 2, which means it has at least two leaves (the 2, plus whatever else) — so it's composite, unless the "whatever else" is empty. That only happens for 2 itself: 2 = 2 , a single leaf. Every other even number is automatically composite. 3. Divisibility, Factors, and Building Up GCF/LCM A factor (or divisor) of n divides evenly into it; a multiple of n is what you get multiplying n by an integer. 3 is a factor of 12 because ; 12 is a multiple of 3 because . The factor tree also finds the Greatest Common Factor and Least Common Multiple of two numbers directly — no listing required. Compare and :

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