Bunuel wrote:
If a, b, c, d, e, f, g, h, i and j are distinct positive integers, what is the units digit of the lowest possible value of \(a^b * c^d * e^f * g^h * i^j\)?
(A) 0
(B) 1
(C) 2
(D) 4
(E) 8
I got Answer A.
If we want the lowest possible value,
pushpitkc , I think it's exactly the reverse of your method (which is where I started, and then changed my mind).
Exponential growth is huge. So I paired largest base with smallest exponent, second largest base with second smallest exponent, and so on.
We need the big factors to be multiplied the fewest times.
Think of 10\(^2\) = 100 compared to 2\(^{10}\) = 1,024.
We can only use the smallest first ten digits once each. So I got
10\(^1\)
9\(^2\)
8\(^3\)
7\(^4\)
6\(^5\)
Where a=10, b=1, c=9, d=2, e=8, f=3, g=7, h=4, i=6, j=5
We can pay attention to just the units' digits of these factors of what I am hoping is the lowest value possible for those digits in that expression.
10\(^1\) = units digit 0
9\(^2\) = units digit 1
8\(^3\) = units digit 2
7\(^4\) = units digit 1
6\(^5\) = units digit 6
0*1*2*1*6 = 0
Let's compare the factors, rounded to the nearest hundred
Large Base Small Exponent:
10\(^1\) = 10
9\(^2\) = 80
8\(^3\) = 500
7\(^4\) = 2,400
6\(^5\) = 8,000
Small Base Large Exponent:
1\(^{10}\) = 1
2\(^9\) = 500
3\(^8\) = 6500
4\(^7\) = 16,000
5\(^6\) = 15,600
The number of zeros in the second group already tells us that that number is larger.
But take just the
first three, then the
last two, factors in each list and multiply
Large Base Small Exponent:
10 * 80 * 500 =
400,000 multiplied by
2,400 * 8,000 =
19,200,000Small Base Large Exponent:
1 * 500 * 6,500 =
3,250,000multiplied by
15,600 * 16,000 =
249,000,000The first set of factors, multiplied, produces a much smaller value than the second set.
There's no way to test those numbers on the exam. But I tested 2\(^{10}\) vs. 10\(^2\) and 3\(^4\) (81) vs. 4\(^3\) (64). They were all calculable. The numbers confirmed my instinct: to get a lower value, multiply the big values as few times as possible. Make the exponents small numbers.
So 10\(^1\) * 9\(^2\) * 8\(^3\) * 7\(^4\) * 6\(^5\) yield units digit:
0*1*2*1*6 = 0
Answer A