It’s Brain Twister week again! Get ready to train your brain.

If a â€“ j are distinct positive integers, what is the units digit of the lowest possible value of a^b x c^d x e^f x g^h x i^j?

(A) 0

(B) 1

(C) 2

(D) 4

(E) 8

Good luck, and don’t forget to check back on Thursday for the answer and explanation. ðŸ™‚

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For lowest possible value the answer is to allocate lowest power value to get in turn least porduct

ie 9^2*8^3*7^4*6^5*1^10. (10^1 not used so as to get the lowest value -1<100)

the answer comes out to be 4.

Hi Katrina,

Your logic seems correct, but the final answer should be (C) 2. What final step did you use to get to 4?

0 is the units digit of the lowest possible value

Lowest possible value = 10^1 * 9^2 * 8^3 * 7^4 * 6^5 * 5^6

Multiply the units digit of each of these pairs, then get units digit of that product

However, 10^1 = 10 –> units digit = 0

0*anything = 0

Therefore, the answer is 0 (without having to multiply to get the actual lowest possible value)

Oops – I added an extra pair!

First sentence should read:

Lowest possible value = 10^1 * 9^2 * 8^3 * 7^4 * 6^5

Hi Christine,

You are very close, except for one little twist :). It should be 1^10 not 10^1, since 1 is less than 10. So, zero will not be amongst any of the units digits. So there’ll actually be a little more grunt work required to get the answer.

Hope that helps!

a – j -> distinct positive integers

#) Lowest possible value of –

a^b x c^d x e^f x g^h x i^j

a=1

b= any positive integer

a^b = 1

c=2

d=3

c^d = 2^3 = 8

e=5

f=4

e^f = 5^4 = 625

g=7

f=6

g^f = 7^6 =117,649

i= 9

j= 8

i^j = 9^8

#) Units Digit

1^any value x 2^3 x 5^4 x 7^6 x 9^8

Units digit – 1 x 8 x 5 x 9 x 1 = 360 = 0

Ans : (A) = 0

Hi Jenil,

Your works with the units digit is correct, but you want to revisit how you arranged the variables. There’s a little logic twist here (remember the question is asking for the “least integer”).

Hope that helps ðŸ™‚

6

Hmm…how did you get ‘6’? Can you show the steps you took to get there? ðŸ™‚

I think the lowest possible value is 1^10*9^2*8^3*7^4*6^5.

I don’t know where I would go from here to get units digit besides doing the calculation.

Is there an easier method?

Hi Keveri,

That is correct in terms of the way the variables should be arranged (points for logic :))

As for the unit digits, you only have to consider the last number. For instance, 9^2 = 1 and 6^5 = 5. Multiply the results of those five pairs of numbers, focusing on the units and that’ll get you the right answer ðŸ™‚

That makes sense! However, I think you meant to say that the units digit of 6^5 = 6

And since 6^2=36, this gives a unit digit of 6. So for any power that 6 is raised to,the units digit will always be 6.

For example,

6^3= 36 * 6

6^4= _ _ 6 * 6

Thanks for the informative answer Chris.

-Keveri

Yes, that’s correct with 6^x always ending in a ‘6’. Sorry for any confusion ðŸ™‚