Quantitative Reasoning – Practice – Find out the unit’s digit of the expression!

units digit

PSA_LEVEL-UP_QNR16. What is the digit at unit’s place of 7777^3333 – 3333^7777?
A) 6
B) 8
C) 4
D) 0

Ans: To find the unit’s digit of 7777^3333 – 3333^7777, we would need to find out the unit’s digit of 7777^3333 and 3333^7777 individually first.

We must notice that Unit’s digit of any number abc^n is same as unit’s digit of c^n as only ‘c’ (the digit at unit’s place) is going to contribute to the unit’s digit of the value of abc^n.

Thus, unit’s digit of 7777^3333 – 3333^7777
=Unit digit of 7^3333 – 3^7777

How to find unit’s digit of 7^3333:

First step is to find out the ‘cyclicity‘ of 7. To do that let us check the pattern shown by powers of 7- the below diagram illustrates that:

units digit2
If you closely observe the pattern, you would find that the unit’s digit repeats itself after every 4 terms; we therefore know the cyclicity of 7 is 4.

units digit3

Next step is to divide the power by the ‘cyclicity’ to get the remainder.

In this case, dividing 3333 by 4 gets you the remainder 1.

Thus the number is of the format 7^(4k+1). Unit’s digit of 7^4k will always be 1.

Thus, unit’s digit of 7^(4k+1) = [unit’s digit of 7^4k]* [unit’s digit of 7^1] = 1*7 =7

Similarly we can find out the unit digit of 3^7777:

Cyclicity of 3 is also ‘4’ as illustrated below:

units digit4

Divide 7777 by 4 and get the remainder; here also remainder is 1. Thus the number is of the format 3^(4k+1).
Thus, unit’s digit of 3^(4k+1) = [unit’s digit of 3^4k] * [unit’s digit of 3^1] = 1*3 =3.

Unit digit of 7^3333 – 3^7777 = 7-3=4.  Option D.

Summarily, these are the steps!!!

units digit6JPG

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