kotela wrote:
Hey friends i have minor doubt can you please clarify this?
What is the units digit of 4^100?
->To my knowledge we generally go with the the number to the power of units digit
Eg:3^84----->since the it follows a sequence such as 3,9,27,81,243......so the units digits follows a sequence 3,9,7,1,3..........so the units digit of 3^84 should be 1....
Now look at the question i.e units digit of 4^100
since units digit sequence of 4 varies in the form 4,6,4........but here the question says 4^0(as the units digit is 0)
so my doubt that are we supposed to consider 4^0 if this is the case then units digit is 1....But Mnhattan Gmat says the units digit of 4^100 is 4...How come this is true?
Its in page 58 Manhattan 12th problem (03 - The Equations, Inequalities, and VICs Guide 4th edition)
Can anyone please explain??
The unit's digit of the base is the one we are focusing on. Not the unit's digit of the power.
Say, you have \(264^{100}\) and you need its unit's digit. How will you approach it?
The unit's digit will be determined by the unit's digit of the base i.e. 4.
Next, the power is 100.
The cyclicity of 4 is 2 i.e. the unit's digit of 4 varies as: 4, 6, 4, 6, 4, 6...
If 4 has an odd power (say \(4^1 or 4^3 or 4^{257}\)), the unit's digit will be 4
If 4 has an even power ((say \(4^2 or 4^8 or 4^{256}\)), the unit's digit will be 6.
100 is even so \(4^{100} = 6\)