Last-Two-Digits Test for 4

Problem Statement

Prove that a number is divisible by 4 if and only if the last two digits form a number divisible by 4.

Hints

Attempt the proof independently first. Focus on the key theorem, algebraic transformation, or contradiction setup that links the hypothesis to the target conclusion.

Solutions

Biconditional Proof (If and Only If)

This requires proving both directions:

Forward Direction ( $\implies$ ): If $N$ is divisible by 4, then its last two digits form a number divisible by 4.

Reverse Direction ( $\impliedby$ ): If the last two digits form a number divisible by 4, then $N$ is divisible by 4.

Setup: Express any integer $N$ as:

N = 100A + L

where $A$ is the number formed by all digits except the last two, and $L$ is the two-digit number formed by the last two digits ( $0 \leq L \leq 99$ ).

Forward Direction Proof:

Assume $4 \mid N$ , so $N \equiv 0 \pmod{4}$ .

Then:

\begin{aligned} 100A + L &\equiv 0 \pmod{4} \end{aligned}

Since $100 = 4 \times 25$ , we have $100A \equiv 0 \pmod{4}$ .

Therefore:

\begin{aligned} 0 + L &\equiv 0 \pmod{4} \\ L &\equiv 0 \pmod{4} \end{aligned}

Thus $4 \mid L$ . \hfill $\square$

Reverse Direction Proof:

Assume $4 \mid L$ , so $L \equiv 0 \pmod{4}$ .

Since $100 = 4 \times 25$ , we have $100A \equiv 0 \pmod{4}$ .

Therefore:

\begin{aligned} N = 100A + L &\equiv 0 + 0 \pmod{4} \\ N &\equiv 0 \pmod{4} \end{aligned}

Thus $4 \mid N$ . \hfill $\square$

Conclusion:

Since both directions are proven, we conclude:

N \text{ is divisible by } 4 \iff \text{last two digits of } N \text{ divisible by } 4

Takeaways

Iff Proof Structure: Must prove both ( $\implies$ ) and ( $\impliedby$ ) directions independently
Digit Representation: Writing $N = 100A + L$ separates last two digits for modular analysis
Key Observation: Since $100 \equiv 0 \pmod{4}$ , divisibility of $N$ by 4 depends only on last two digits
Generalization: Same technique proves divisibility rules for powers of 2 (e.g., rule for 8 uses last three digits)

Further Readings

If you found this proof interesting, be sure to check out these relevant HSC booklets to sharpen your reasoning skills:

HSC Proofs: https://vumaths.com/booklets/hsc-proofs/
HSC Vectors: https://vumaths.com/booklets/hsc-vectors/
HSC Mechanics: https://vumaths.com/booklets/hsc-mechanics/

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