What's the units digit Questions?

Question

I can't count the number of questions I've come across of this nature:

What is the units digit of (9^3)(4^5)(6^3)?

any ideas as to how I can solve questions like this?

abhijit_sen · Answer

(9^3)(4^5)(6^3) can be re-written as
=(9*4*6)^3 * 4^2
= (216)^3 * 16

As 6^3 = 216 (6 cube ends in 6, so 216 cube will also end in 6)

xxxxxxxx6 * 16

Will end in 6.

ocprep · Answer

1) 9*9*9 = 72 9 
2) 44444 = 102 4 
3) 666 = 21 6

basically you take the units digit from each of numbers which is...9,4,6 then multiply those numbers
9*4*6 = 21 6

Answer is 6. 
Anyone know a faster way?

streamingline · Answer

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

Thanks much

soods26 · Answer

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.

rajesh1088 · Answer

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

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Thanks Rajesh

What's the units digit Questions?

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