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Intern  Joined: 14 Mar 2013
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What is the units digit of the expression 14^7−18^4?  [#permalink]

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Question Stats: 75% (01:27) correct 25% (01:43) wrong based on 318 sessions

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What is the units digit of the expression 14^7−18^4?

(A) 0
(B) 3
(C) 4
(D) 6
(E) 8

Originally posted by amgelcer on 19 Oct 2013, 14:49.
Last edited by Bunuel on 20 Mar 2014, 08:03, edited 2 times in total.
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Re: What is the units digit of the expression 14^7−18^4?  [#permalink]

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I love this type of question. It looks scary until you know how to handle it.

Start by finding the units digit of each of the components:

14^7 --> all that matters is the 4, so figure out the pattern 4^1=...4 4^2=...6 4^3=...4, so 4^7=...4 (odd power)

18^4 --> same approach. 8^1=...8 8^2=...4 8^3=...2 8^4=...6. (Not quite confirming the pattern, but we got what we needed)

So, we have

...4
-...6
_____
Borrow a 1, and we have
...8

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Re: What is the units digit of the expression 14^7−18^4?  [#permalink]

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Side note on this one. Make sure that the result isn't a negative number, since that can change the outcome.

Thankfully, here we shouldn't have to calculate large numbers in order to confirm this, but it's certainly worth taking a second look!
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Re: What is the units digit of the expression 14^7−18^4?  [#permalink]

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1
14^7−18^4 = 2^7*7^7 - 2^4*9^4
= 2^4 (2^3*7^7 - 9^4)
=16[ 8* (unit digit 3) - unit digit 1 ]
= 16 (unit digit 4 - unit digit 1)
= 16 *unit digit 3
= unit digit 8

unit digit = 8
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Re: What is the units digit of the expression 14^7−18^4?  [#permalink]

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1
First expression = 14^7 it cycles around 4,6,4,6............ seventh power would have 4 in units place
Second expression = 18^4 it cycles around 8,4,2,6,............ seventh power would have 6 in units place

14 - 6 = 8 (1 has to be borrowed from tens place as 6>4)

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Re: What is the units digit of the expression 14^7−18^4?  [#permalink]

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I think answer on this one should be E too. Since we know that 14^7>18^4, as Will said one should always check if the number is positive.

Is the OA shown as D correct for any reason?

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Re: What is the units digit of the expression 14^7−18^4?  [#permalink]

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jlgdr wrote:
I think answer on this one should be E too. Since we know that 14^7>18^4, as Will said one should always check if the number is positive.

Is the OA shown as D correct for any reason?

Cheers
J

The OA is E. Thank you. Edited.

Units digits, exponents, remainders problems to practice: new-units-digits-exponents-remainders-problems-168569.html

Hope it helps.
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Re: What is the units digit of the expression 14^7−18^4?  [#permalink]

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amgelcer wrote:
What is the units digit of the expression 14^7−18^4?

(A) 0
(B) 3
(C) 4
(D) 6
(E) 8

you have to use pattern method.
powers of 4 ends with unit digits: 4,6,4,6 and so on); if the power is odd it is 4 otherwise 6
powers of 8 ends with unit digits: 8,4,2,6,8,4,2,6 and so on); it is cycle of four
so the unit digit is 4-6=8
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Re: What is the units digit of the expression 14^7−18^4?  [#permalink]

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Hi,

In the example mentioned above, does anyone know why do we borrow 1, make 14 and subtract 6 from it to get 8 as the answer? What about if the questions were "Whats the unit digit of 18^4 - 14^7?" In that case the unit digit would have been 6 - 4 = 2 OR?

Thanks for your help!

WillEconomistGMAT wrote:
I love this type of question. It looks scary until you know how to handle it.

Start by finding the units digit of each of the components:

14^7 --> all that matters is the 4, so figure out the pattern 4^1=...4 4^2=...6 4^3=...4, so 4^7=...4 (odd power)

18^4 --> same approach. 8^1=...8 8^2=...4 8^3=...2 8^4=...6. (Not quite confirming the pattern, but we got what we needed)

So, we have

...4
-...6
_____
Borrow a 1, and we have
...8
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Re: What is the units digit of the expression 14^7−18^4?  [#permalink]

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Units digit of 14^7−18^4:
Consider the following unit digits:
4^1: 4
4^2: 6
4^3: 4
4^4: 6
⇒ unit digit of 4^7 is: 4
The unit digit of 14^7 will also be 4

Also:
8^1: 8
8^2: 4
8^3: 2
8^4: 6
⇒ unit digit of 8^4 is: 6
The unit digit of 18^4 will also be 6

Finally, 4-6 (or 14-6) = 8
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Re: What is the units digit of the expression 14^7−18^4?  [#permalink]

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amgelcer wrote:
What is the units digit of the expression 14^7−18^4?

(A) 0
(B) 3
(C) 4
(D) 6
(E) 8

Cyclicity of units digit of 4 is 2

So, 4^7 will have units digit as 4

Units digit of 8^4 is 6

Now, 4 - 6 = 8

Hence, units digit will be (E) 8

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Re: What is the units digit of the expression 14^7−18^4?  [#permalink]

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amgelcer wrote:
What is the units digit of the expression 14^7−18^4?

(A) 0
(B) 3
(C) 4
(D) 6
(E) 8

Since we only care about units digits, we can rewrite the expression as:

4^7 − 8^4

Let’s start by evaluating the pattern of the units digits of 4^n for positive integer values of n. That is, let’s look at the pattern of the units digits of powers of 4. When writing out the pattern, notice that we are ONLY concerned with the units digit of 4 raised to each power.

4^1 = 4

4^2 = 6

4^3 = 4

4^4 = 6

The pattern of the units digit of powers of 4 repeats every 2 exponents. The pattern is 4–6. In this pattern, all positive exponents that are odd will produce a 4 as its units digit. Thus:

4^7 has a units digit of 4.

Next, we can evaluate the pattern of the units digits of 8^n for positive integer values of n. That is, let’s look at the pattern of the units digits of powers of 8. When writing out the pattern, notice that we are ONLY concerned with the units digit of 8 raised to each power.

8^1 = 8

8^2 = 4

8^3 = 2

8^4 = 6

8^5 = 8

The pattern of the units digit of powers of 8 repeats every 4 exponents. The pattern is 8–4–2–6. In this pattern, all positive exponents that are multiples of 4 will produce an 8 as its units digit. Thus:

8^4 has a units digit of 6.

Thus, the “units digit” of 4^7 − 8^4 is 4 − 6 = –2. However, we can’t have a negative number as the unit digit. When we encounter such a case, we add 10 to make it positive. Thus, the the units digit of 4^7 − 8^4 is –2 + 10 = 8.

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