# Short Cut #1: The Magic Remainder Technique

What is the units digit (last digit) of 22011 ?

THE MAGIC REMAINDER TECHNIQUE

(For numbers with a very large exponents)

To get the last digit of a number with a very large exponent:

Step 1.  Get last 2 digits of the exponent.

Step 2.  Divide by 4 and get remainder.

Step 3.  The remainder becomes the new exponent.

Example:

• What is the units digit (last digit) of 22011 ?

Step 1.  22011  –> 211

Step 2.  Divide 11 by 4.  Remainder = 3

Step 3.  211 –>  23 = 8

Exception to the Rule:  If the remainder is zero, exponent should be 4, not 0.

Example:   312 –> 34 = 81.  Last digit = 1.

1. What is the units digit of 32014 ?
2. What is the units digit of 7326 ?
3. What is the units digit of 9878 ?
4. What is the units digit of 62015 ?
5. CHALLENGE:  What is the units digit of 1232014 ?

Reflection for the Pro (if you are BRAVE enough!)

1.     If the base has more than one digit, what shortcut can you suggest?
2.     Think of a faster way if the base is 4 or 9. What if the base is 1, 5, 6 or 0?

–Wacks Formula 🙂

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ShortCut #2: The Wacks Middle Finger Technique

Disclaimer:  The name of the formula is purely for fun, and does not discredit others who might have discovered the same formula independently.