General number properties

[GmatTakerNo.1]

I have absolutely no idea how to solve this in two minutes.

Can you explain me one approach, please?

What is the units digit of 3^17×17^54×135^19×31^21×42^17?

The answer is 0.

Thank you very much.

[akhpad]

You need to know power cycle

power cycle of 2 is 2,4,8,6
power cycle of 3 is 3,9,7,1
power cycle of 5 is 5
power cycle of 7 is 7,9,3,1

I will take only UNIT digit
3^17X17^54X135^19X31^21X42^17?

= 3 * 9 * 5 * 2 = 270 (Unit digit is ZERO); it is the way to find unit digit only.

[tpr-becky]

The units digit of any number is obtained by multiplying the units digits - you don't have to worry about all the other digits.

The 135^19 will always end in a 5 because a 5 to an exponent always ends in a 5. the 42^17 will end in a 2 becuase the units digits of 2 to an exponent have a pattern of 2, 4, 8, 6. if you multiply 5(2) you will always end up with a units digit and it won't matter what other numbers you multiply by.

This falls under the concept of when the math gets too hard there is a pattern - find it.

[GmatTakerNo.1]

Thank you guys.

[harshavardhanc]

you just need a 5 and a 2 to decide the units digit, as it will be always be 0 if you find these two numbers in a product.

135 will give you a 5 and 42 will give you a two. That's it. therefore, the units digit of this messy product will be 0.