Units Digit – Core Concept

The units digit is the last digit before the decimal point.

Examples:

• Units digit of 79 is 9
• Units digit of 76546 is 6

Useful in:

• Multiplication of large numbers
• Powers and exponentiation
• Modulo 10 and 5 operations

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Units Digit in Multiplication

To simplify: take only the units digit of each number.

Example:

What is the units digit of
85945 \times 89 \times 58307?

Use only last digits:
5 \times 9 \times 7 = 45 \times 7 = 315 → units digit = 5

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Cyclicity of Units Digits

The units digit of powers repeats in cycles. Identify the cycle, divide the exponent by the cycle length, and use the remainder to find the result.

Cyclicity Table

Cycle Length	Digits	Example Powers
1	0, 1, 5, 6	Always same (e.g., 5^n = 5)
2	4, 9	4^1 = 4, 4^2 = 6, repeat
4	2, 3, 7, 8	Repeats every 4 (see below)

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Example: 7^{345}

Cycle of 7:

\begin{align*} 7^1 &= 7 \ 7^2 &= 49 \Rightarrow 9 \ 7^3 &= 343 \Rightarrow 3 \ 7^4 &= 2401 \Rightarrow 1 \end{align*}

Cycle: 7, 9, 3, 1
Now,
345 \mod 4 = 1 ⇒ Use first value in cycle → 7

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Summary of Exponent Rules

Type	Rule
1-cycle	Units digit = base
2-cycle	Odd exponent → base, even exponent → base²
4-cycle	Divide exponent by 4, use remainder to pick from cycle

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Factorials – Trailing Zeros & Prime Powers

Factorials are common in GMAT questions involving:

• Trailing zeros
• Power of a prime
• Divisibility by a composite number

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Trailing Zeros in n!

Each trailing zero = factor of 10 = 2 \times 5
Since 2s are plenty, just count 5s:

Formula:

\left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + \cdots

Example: Zeros in 32!

\left\lfloor \frac{32}{5} \right\rfloor + \left\lfloor \frac{32}{25} \right\rfloor = 6 + 1 = \boxed{7}

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Power of a Prime in n!

Formula:

\sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor

Stop when p^k > n

Example: Power of 2 in 25!

\left\lfloor \frac{25}{2} \right\rfloor + \left\lfloor \frac{25}{4} \right\rfloor + \left\lfloor \frac{25}{8} \right\rfloor + \left\lfloor \frac{25}{16} \right\rfloor = 12 + 6 + 3 + 1 = \boxed{22}

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Power of Non-Prime in n!

Steps:

1. Prime factorize the number
2. Find power of each prime in n!
3. Divide each result by exponent in factorization
4. Take the minimum

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Example: Power of 900 in 50!

Factor: 900 = 2^2 \times 3^2 \times 5^2

Step 1: Count powers in 50!

Prime	Calculation	Result
2	25 + 12 + 6 + 3 + 1	47
3	16 + 5 + 1	22
5	10 + 2	12

Step 2: Divide by exponent in 900

Prime	Divide by	Final Power
2	2	23
3	2	11
5	2	6

Final Answer:
900^6 divides 50!

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Manual vs Formula Check: 17!

Prime	Formula	Total
2	8 + 4 + 2 + 1 = \boxed{15}	15
3	5 + 1 = \boxed{6}	6
5	3 = \boxed{3}	3

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GMAT Tricks Summary

What You’re Finding	Focus On
Trailing Zeros	Count 5s only
Number of 15s in n!	Count 5s (3s are more frequent)
Number of 10s in n!	Count 5s
Number of 8s = 2^3 in n!	Count 2s, then divide by 3

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Final Takeaway

Use these tools to simplify and speed up:

• Units digit: Use cyclicity rules and remainder logic
• Factorials: Use prime count formulas and shortcuts
• Apply patterns: Recognize them quickly and eliminate wrong choices fast