What is the unit digit of the sum...

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What is the unit digit of the sum...

by BTGmoderatorLU » Tue Oct 24, 2017 1:52 pm
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.

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by ceilidh.erickson » Wed Oct 25, 2017 3:24 pm
The units digits of a given number raised various exponents will always form a pattern:

2^1 = 2
2^2= 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512

As we can see, the pattern repeats after every 4th one. Every exponent that's a multiple of 4 will have a units digit of 6. We can extrapolate the rule:
2^(4n) ---> ends in 6
2^(4n + 1) ---> ends in 2
2^(4n + 2) ---> ends in 4
2^(4n + 3) ---> ends in 8

This will be true for every digit. For this problem, we need the patterns for 3's, 5's, and 2's:
3^1 = 3
3^2 = 9
3^3 = 7 (only worry about the UD)
3^4 = 1
3^5 = 3
3^6 = 9
Repeats every 4.

pattern for 5's:
5^1 = 5
5^2 = 5 (only worry about UD)
Pattern: the units digit will always be 5.

3^47 ---> 3^(44 + 3) ---> 3^(4n + 3) ---> UD of 7
5^43 ---> doesn't matter what the exponent is. UD = 5
2^12 ---> 2^(4n) ---> UD of 6

When we add 3 large numbers together and we're looking for the UD, we can ignore everything except the UD of those 3 numbers. 7 + 5 + 6 = 18, so the UD = 8.

The answer is A.
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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.
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by [email protected] » Wed Nov 20, 2019 5:44 pm
BTGmoderatorLU wrote: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.
The units digit of the sum 3^47 + 5^43 + 2^12 is the sum of the units digits of 3^47, 5^43, and 2^12 (if this sum is more than 10, then the units digit is the remainder when this sum is divided by 10).

Since the pattern of units digits of the powers of 3 is 3-9-7-1, then 3 raised to an exponent that is a multiple of 4 leaves a units digit of 1. Thus, 3^48 has a units digit of 1, so 3^47 has a units digit of 7.

5 raised to any positive integer exponent will always have a units digit of 5.

Since the pattern of units digits of the powers of 2 is 2-4-8-6, then 2 raised to an exponent that is a multiple of 4 leaves a units digit of 6. Thus, 2^12 has a units digit of 6.

Since the sum of the units digits is 7 + 5 + 6 = 18, and 18/10 = 1 R 8, the units digit of the given sum is 8.

Answer: A

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