What's the units digit Questions?

Hi Bunuel, could you please suggest your way of attacking this type of question?

Thanks much

In these type of questions it helps to remember that all number from 1-9 when raised to a power have a cyclicity for the unit digits.

For example taking 4 for instance : 4,16,64,256.... As seen, powers of 4 repeat their unit digits at an interval of 2.

Similarly for 9 : 9,81,729,..... powers of 9 Also repeats their units digit at an interval of 2.

For 6: 6,36,216..... The units digit of powers of 6 remain the same...

Thus, for the question we get that the units digit of 9^3 = 9, 4^5 = 4, 6^3 = 6.

The multiplication of 9,4, as units digit would give 36 as the units digit and subsequent multiplication with a number with unit's digit 6 will also give 6 as the unit's digit.

TIP: Try to memorize the intervals at which the exponents of numbers from 1-9 repeat their unit's digits.

The best way to tackle such questions is following the rule of cyclicity:

consider digit 2: \(2^1= 2\) \(2^2= 4\) \(2^3= 8\) \(2^4= 16\) \(2^5= 32\) .. take a close look at \(2^5\) the units digit repeats itself. Therefore 2 has a cyclicity of 4. Similarly digits 3,7 and 8 also have a cyclicity of 4.

digits 4 and 9: cyclicity of 2, and digits 5 and 6: units digit same as number itself.

coming to the qn: (9^3)(4^5)(6^3)

Ans: 9^3: 9 has cyclicity of 2, therefore divide its index i.e 3 by 2: you get remainder as 1: Therefore units digit is 9^1 i.e 9.
4^5: 4 has cyclicity of 2, therefore divide its index i.e 5 by 2: you get remainder as 1: Therefore units digit is 4^1 i.e 4.
6^3: Units digit will be 6 itself as explained in the theory above.

Now to get the units digit of the entire expression multiply the results above : 9*4*6= 36*6= units digit of 6.

Hope my explanation is clear. Once you get hold of this method, units digit can be calculated in seconds.

Thanks
Rajesh