Factors of a number are the whole numbers that divide it EVENLY (no remainder). Multiples of a number are what you get when you multiply it by 1, 2, 3, ...
Factors of 12
→1, 2, 3, 4, 6, 12
Multiples of 12
→12, 24, 36, 48, 60, ...
4.E.1 — Factors and multiples (1–100).
Step 01 of 04
Factors of a number are the whole numbers that divide it EVENLY (no remainder). Multiples of a number are what you get when you multiply it by 1, 2, 3, ...
Factors of 12
→1, 2, 3, 4, 6, 12
Multiples of 12
→12, 24, 36, 48, 60, ...
Step 02 of 04
Find ALL factors by checking 1, 2, 3, ... up to the square root.
Factors of 24: 1×24, 2×12, 3×8, 4×6. Stop when you reach a repeat (5×?; 5 doesn't divide; next would be 6×4 — already have).
factors of 24={1,2,3,4,6,8,12,24}
Step 03 of 04
Multiples are infinite. List the first few — they're just the times-table.
multiples of 7=7,14,21,28,35,42,49,56,63,70,…
Step 04 of 04
Common factors and multiples.
Common factor (of 12 and 18)
→a number that divides both — 1, 2, 3, 6
Greatest common factor (GCF)
→6
Common multiple (of 4 and 6)
→a number that's a multiple of both — 12, 24, 36, ...
Least common multiple (LCM)
→12
Key insight
Factors divide INTO the number; multiples are what you get when you multiply OUT. Factors are finite (and usually few); multiples go on forever. GCF and LCM compare two numbers' lists.
4.E.2 — Prime vs. composite numbers.
Step 01 of 04
Prime number = a whole number greater than 1 with EXACTLY two factors: 1 and itself.
Composite = greater than 1 with MORE than two factors.
1
→NEITHER prime nor composite (only one factor)
2
→PRIME (smallest, only even prime)
3, 5, 7, 11
→PRIME
4, 6, 8, 9, 10
→COMPOSITE
Step 02 of 04
Primes under 30 — memorize.
2,3,5,7,11,13,17,19,23,29
Notice: 2 is the only even prime. Every other even number is composite (divisible by 2).
Step 03 of 04
How to test if a number is prime.
Try dividing by every prime up to (and including) the SQUARE ROOT of the number. If none divide evenly, it's prime.
Test 47: 47≈6.9, so check primes 2, 3, 5. None divide 47 evenly → 47 is PRIME.
Step 04 of 04
Quick divisibility checks.
Divisible by 2
→last digit even
Divisible by 3
→digit sum divisible by 3
Divisible by 5
→last digit is 0 or 5
Divisible by 9
→digit sum divisible by 9
Divisible by 10
→last digit is 0
Key insight
Prime = exactly two factors (1 and itself). 1 is neither. Test by dividing by primes up to the square root. Memorize divisibility tricks to speed it up.
4.E.3 — Number and shape patterns from a rule.
Step 01 of 04
A pattern follows a rule that produces each next term from the previous one (or from its position).
Add 5 each time
→3, 8, 13, 18, 23, ...
Subtract 4
→40, 36, 32, 28, ...
Double
→2, 4, 8, 16, 32, ...
Multiply by 3
→1, 3, 9, 27, 81, ...
Step 02 of 04
Find the rule from a sequence. Compare consecutive terms.
5, 11, 17, 23, ... — differences are 6, 6, 6 — rule is "add 6." Next two: 29, 35.
2, 6, 18, 54, ... — ratios are 3, 3, 3 — rule is "multiply by 3." Next two: 162, 486.
Step 03 of 04
Combined-rule patterns. Sometimes the rule uses both addition AND multiplication.
1, 3, 7, 15, 31, ... Not constant differences (2, 4, 8, 16). Each new difference doubles. Rule: "double the previous term, then add 1." Verify: 31 × 2 + 1 = 63.
Step 04 of 04
Shape patterns. Same idea but with figures.
1 dot, 3 dots, 6 dots, 10 dots (triangular numbers). Differences: 2, 3, 4 — rule is "add the next counting number." Next: 15, then 21.
Key insight
Look at differences (or ratios) between consecutive terms to spot the rule. Add same → arithmetic. Multiply same → geometric. More complex rules combine operations. Always check the rule against several pairs before extending the pattern.
Free diagnostic
Test your understanding: 10 questions, ~10 min.
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