Page 1 of 3
💡 If-Then Statements
A direct proof works by connecting facts in a chain:
"IF something is true, THEN something else must also be true."
By linking several if-then statements together, you can prove things step by step — like building a bridge from what you know to what you want to show.
Example 1: If a number is divisible by 10, then it's divisible by 5
START
Let's say a number is divisible by 10.
⬇️
REASON
That means it equals 10 × (something). For example, 30 = 10 × 3.
⬇️
REASON
But 10 = 5 × 2, so 10 × (something) = 5 × 2 × (something).
⬇️
PROVED!
This means the number is 5 × (something), so it's divisible by 5. ✓
Each step follows logically from the one before it. That's a proof!
Example 2: If a number is divisible by 6, then it's divisible by 3
START
The number is divisible by 6.
⬇️
REASON
So it equals 6 × (something).
⬇️
REASON
6 = 3 × 2, so 6 × (something) = 3 × 2 × (something).
⬇️
PROVED!
The number is 3 × (something), so it's divisible by 3. ✓
Page 2 of 3
1 Complete the If-Then Chain 🔗
Fill in the missing steps to complete each proof.
Prove: If a number is divisible by 12, then it's divisible by 4.
START The number is divisible by 12.
⬇️
REASON So it equals 12 × (something).
⬇️
REASON 12 = 4 × , so 12 × (something) =
⬇️
PROVED!
Prove: If a whole number ends in 0, then it's even.
START The number ends in 0.
⬇️
REASON Numbers ending in 0 are divisible by .
⬇️
REASON 10 = 2 × , so the number is divisible by .
⬇️
PROVED!
2 Match If-Then Pairs 🔀
Draw a line connecting each IF to the correct THEN.
IF a number is divisible by 100…
THEN it ends in 5 or 0.
IF a number is divisible by 9…
THEN it is divisible by 50.
IF a number is a multiple of 4…
THEN its digits add up to a multiple of 9.
IF a number is divisible by 5…
THEN it is even.
Page 3 of 3
3 Write Your Own Proof! ✍️
Choose one claim and write a step-by-step proof using if-then reasoning.
Choose one to prove:
A) If a number is divisible by 8, then it's divisible by 2.
B) If a number is a multiple of 15, then it's a multiple of 5.
C) If you add any two multiples of 3, the result is also a multiple of 3.
I chose:
START
⬇️
REASON
⬇️
REASON
⬇️
PROVED!
🌟 Bonus: Prove a second claim!
Pick a different letter (A, B, or C) from above and prove it here too.
I chose:
START
⬇️
REASON
⬇️
PROVED!
🎓 Congratulations! You've learned 5 proof techniques:
checking examples vs. proof,
counterexamples,
visual/dot proofs,
picture proofs,
proof by cases, and
if-then reasoning. You're thinking like a real mathematician!
Answer Key:
1a) 12=4×3, 4×3×(something), divisible by 4 — 1b) divisible by 10, 10=2×5, divisible by 2, number is even —
2) div by 100→div by 50; div by 9→digits add to mult of 9; mult of 4→even; div by 5→ends in 5 or 0 —
3) A: 8=2×4, so 8×n=2×4×n, divisible by 2; B: 15=5×3, so 15×n=5×3×n, divisible by 5; C: 3a+3b=3(a+b), multiple of 3