Number Theory

Key points

• Divisibility. By 3 or 9: add the digits. By 4: check the last two digits. By 8: the last three. By 6: it must pass both 2 and 3. By 11: add and subtract the digits in turn; the result must be 0 or a multiple of 11.
• Factors come in pairs. 36 = 1 × 36 = 2 × 18 = 3 × 12 = 4 × 9 = 6 × 6. Stop checking once the pair meets in the middle.
• Primes have exactly two factors, 1 and themselves. 1 is neither prime nor composite, and 2 is the only even prime. There are 25 primes below 100.
• Exponents. 2³ means 2 × 2 × 2 = 8, not 2 × 3 = 6.
• Counting choices. 4 meals and 3 drinks make 4 × 3 = 12 pairs. Multiply the choices; adding them (7) is the trap.
• How many factors? Write the number as primes with exponents, add 1 to each exponent, then multiply. 72 = 2³ × 3², so it has (3 + 1)(2 + 1) = 12 factors. The extra 1 is the choice of using none of that prime.
• Counting multiples. From 1 to 100 there are 33 multiples of 3, 20 multiples of 5, and 6 multiples of both (multiples of 15). Multiples of 3 or 5: 33 + 20 − 6 = 47. Subtract the overlap so nothing is counted twice.
• GCF or LCM? Splitting into equal groups with nothing left over calls for the GCF. Things that repeat and meet again call for the LCM. GCF × LCM equals the product of the two numbers: for 12 and 18, 6 × 36 = 12 × 18 = 216.
• Units digits repeat. Powers of 3 end in 3, 9, 7, 1, then the cycle starts again.
• Roman numerals. I 1, V 5, X 10, L 50, C 100, D 500, M 1000. A smaller numeral before a larger one means subtract: IV is 4, XC is 90, CM is 900.