Numbers
The Numbers domain tests fluency with the real number system: order of operations, signed arithmetic, absolute value, factors and multiples, primes, fractions, exponents, radicals, scientific notation, ratios, and unit conversion. These questions are the most computational on the exam, so most errors come from sign handling and from skipped conversion steps.
Order of operations and signed numbers
- Parentheses first, then exponents, then multiplication and division left to right, then addition and subtraction left to right.
- Watch the notation: −2² = −4 because the exponent applies before the negative sign, but (−2)² = 4.
- Absolute value is distance from zero, so |x| = 5 means x = 5 or x = −5.
Factors, multiples, and primes
- Divisibility tests: by 3 or 9 use the digit sum; by 4 use the last two digits; by 6 require divisibility by both 2 and 3.
- 1 is neither prime nor composite, and 2 is the only even prime.
- Prime factorization gives the GCD from the lowest power of each shared prime and the LCM from the highest power of every prime appearing.
Fractions, exponents, and radicals
- To divide fractions, multiply by the reciprocal: a/b ÷ c/d = a/b × d/c.
- a^m·a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^0 = 1 for a ≠ 0.
- √(ab) = √a·√b; rationalize a denominator by multiplying by the conjugate.
- Scientific notation is a × 10^n with 1 ≤ |a| < 10.
Worked example
Find the GCD and LCM of 84 and 120. 84 = 2²·3·7 and 120 = 2³·3·5. GCD uses the lowest shared powers: 2²·3 = 12. LCM uses the highest power of every prime: 2³·3·5·7 = 840. Check with the identity GCD × LCM = 12 × 840 = 10,080 = 84 × 120.
Traps
- Mixing up GCD and LCM: the GCD divides both numbers, the LCM is divisible by both.
- Cancelling across an addition sign, for example claiming (2 + 3)/3 = 2.
- Converting units in the wrong direction: 3 m is 300 cm, so multiply by 100 rather than divide.
- Rounding intermediate values and losing precision in the final answer.