What is the Unit’s Digit of \((23)^6(17)^3(61)^9\) GMAT Problem Solving

Question: What is the unit’s digit of\((23)^6(17)^3(61)^9\)?

1. 1
2. 3
3. 5
4. 7
5. 9

Answer:
Approach Solution (1):
The unit digit calculation has nothing to do with any digit of number other than the unit digit only i.e. unit digit of\((2349)^7\)will be same as the unit digit of\((9)^7\)
The cyclicity of unit digit of 3 and 7 are four i.e. the unit digit of any number with unit digits 3 or 7 repeats after every 4th power
i.e. units digit of\((23)^6(17)^3(61)^9\)= units digit of\((3)^6(7)^3(1)^9\)= units digit of\((3)^2(7)^3(1)^9=9*3*1=7\)
Correct option: D

Approach Solution (2):
Do remember this one.
Cyclicity of Numbers Summary
Number. Cyclicity of a number
Cyclicity of 1= 1
Cyclicity of 2 = 4
Cyclicity of 3 = 4
Cyclicity of 4 = 2
Cyclicity of 5 = 1
Cyclicity of. 6 = 1
Cyclicity of 7 = 4
Cyclicity of 8 = 4
Cyclicity of 9 = 2
Cyclicity of 10 = 1
If you don't know how to calculate cyclicity of a number do let me know. I will help out.
Now coming back to question,
23^6
Unit digit = 3
Cyclicity of 3 = 4
Divide power i.e. 6 by 4, We will get Remainder = 2
17^3
Unit digit = 7
Cyclicity of 7 = 4
Divide power i.e. 7 by 4, We will get Remainder = 3
61^9
Unit digit = 1
Cyclicity of 7 = 1
Divide power i.e. 1 by 1, We will get Remainder = 0.
Again coming back to question
So,\((3)^2(7)^3(1)^9=9*3*1=7\)
Correct option: D

Approach Solution (3):
The cyclicity of unit digit of 3 and 7 are four i.e. the unit digit of any number with unit digits 3 or 7 repeats after every 4th power
i.e. units digit of\((23)^6(17)^3(61)^9\)= units digit of\((3)^6(7)^3(1)^9\)= units digit of\((3)^2(7)^3(1)^9=9*3*1=7\)
Correct option: D

“What is the unit’s digit of\((23)^6(17)^3(61)^9\)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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