What is the unit digit of the sum 3^47 + 5^43 + 2^12?
A) 8
B) 4
C) 9
D) 1
E) 3
The OA is A.
Can any expert help me with this PS question please? Thanks.
Hi LUANDATO,
Let's take a look at your question.
We will first look at some of the exponents of 3, 5 and 2 to find out the pattern of units digit in each number.
$$3^1,\ 3^2,\ 3^3,\ 3^4,\ 3^5,\ 3^6,\ ...$$
$$=3,\ 9,\ 27,\ 81,\ 243,\ 729,...$$
We can see that in the powers of 3, the unit digit repeats itself after every 4 numbers in the sequence.
Let's now examine the pattern of 5
$$5^1,\ 5^2,\ 5^3,\ 5^4,\ 5^5,\ 5^6,\ ...$$
$$=5,\ 25,\ 125,\ 625,\ ...$$
Unit digit is 5 for all the exponents of 5.
Let's now examine the pattern of 2
$$2^1,\ 2^2,\ 2^3,\ 2^4,\ 2^5,\ 2^6,\ ...$$
$$=2,\ 4,\ 8,\ 16,\ 32,\ 64,\ ...$$
Now using these patterns we will find out the unit digit of 3^47.
Since unit digit in exponents of 3's sequence repeats after every 4 exponents.
47th power = 4(11) + 3 =>
3rd exponent's unit digit in 3's sequence is 7
For all the exponents of 5 unit digit is 5, so
unit digit of 5^43 is 5.
Now, we will find out the unit digit of 2^12.
Since unit digit in exponents of 2's sequence repeats after every 4 exponents.
12th power = 2(6)+0 =>
last exponent's unit digit in 2's sequence is 6
Let's now add up all the unit digits = 7 + 5 + 6 = 18
Hence, the unit digit is 8.
Therefore, Option
A is correct.
Hope it helps.
I am available if you'd like any follow up.
