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What's the units digit Questions?

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Intern
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What's the units digit Questions? [#permalink] New post 31 Mar 2008, 16:09
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?
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Re: What's the units digit Questions? [#permalink] New post 31 Mar 2008, 16:15
(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.
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Re: What's the units digit Questions? [#permalink] New post 31 Mar 2008, 16:19
1) 9*9*9 = 729
2) 44444 = 1024
3) 666 = 216

basically you take the units digit from each of numbers which is...9,4,6 then multiply those numbers
9*4*6 = 216

Answer is 6.
Anyone know a faster way?
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Re: What's the units digit Questions? [#permalink] New post 03 Feb 2014, 22:05
Liquid wrote:
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?


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

Thanks much
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Re: What's the units digit Questions? [#permalink] New post 03 Feb 2014, 22:28
Liquid wrote:
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?


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.
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Re: What's the units digit Questions? [#permalink] New post 04 Feb 2014, 11:17
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. 8-)

Thanks
Rajesh
Re: What's the units digit Questions?   [#permalink] 04 Feb 2014, 11:17
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