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# What is the units digit of 9^19 - 7^15?

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Math Expert
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What is the units digit of 9^19 - 7^15?  [#permalink]

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07 Oct 2019, 21:20
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Difficulty:

15% (low)

Question Stats:

81% (00:56) correct 19% (01:32) wrong based on 57 sessions

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Competition Mode Question

What is the units digit of $$9^{19} - 7^{15}$$?

A. 2
B. 4
C. 5
D. 6
E. 7

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Re: What is the units digit of 9^19 - 7^15?  [#permalink]

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07 Oct 2019, 21:33
Cyclic order of 9 raised to the power is [1,9]

Cyclic order of 9 raised to the power is [9,3,1,7]

So, the difference of the said expression is 2

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Re: What is the units digit of 9^19 - 7^15?  [#permalink]

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07 Oct 2019, 21:45
1
9^19 - 7^15

9^1 = 9 unit digit = 9
9^2 = 81 unit digit = 1
9^3 = 729 unit digit = 9
9^4= 6561 unit digit = 1
hence cyclisity of the powers of 9 is 2

7^1= 7 unit digit =7
7^2= 49 unit digit =9
7^3= 343 unit digit =3
7^4= 2401 unit digit =1
7^5=16807 unit digit =7
hence cyclisity of the powers of 7 is 4

9^19 - 7^15 = (9^18)*9 - (7^12)*7^3
=1*9 - 1*3 = 9-3 =6

Hence D.
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Re: What is the units digit of 9^19 - 7^15?  [#permalink]

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07 Oct 2019, 22:20
1
What is the units digit of $$9^{19}−7^{15}$$?

A. 2
B. 4
C. 5
D. 6
E. 7

For $$9^{19}$$ unit digit would be 9 as odd power of 9 gives 9 as unit digit(even power gives one)

For $$7^{15}$$ unit power would be 3 since 7 raised to power which is a multiple of 3 gives 3 as unit digit.

Hence unit place of $$9^{19}−7^{15}$$ = Unit place of $$9^{19}$$ - Unit place of $$7^{15}$$
= 9 - 3
= 6

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Re: What is the units digit of 9^19 - 7^15?  [#permalink]

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08 Oct 2019, 02:53
1
1
What is the units digit of 9^19 - 7^15?

9^1=9
9^2=81
9^3= _ _ 9
the pattern is _9,_1,_9,_1
19/2=8 R1
therefore the unit digit of 9^19 is 9

7^1=7
7^2=49
7^3=_ _ 3
7^4= _ _ _ 1
7^5= _ _ _ _ _ 7
the pattern is _7, _9,_3,_1
15/4=3 R3
therefore the unit digit of 7^15 is 3

9-3= 6

D
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Re: What is the units digit of 9^19 - 7^15?  [#permalink]

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08 Oct 2019, 03:52
1
Quote:
What is the units digit of $$9^{19}−7^{15}$$?

A. 2
B. 4
C. 5
D. 6
E. 7

$$units:9^{19}…cycles(9)=[9,1]=2…remainder:19/2=1…units:1st=[9]$$; or,
$$units:9^{19}=(3^2)^{19}=3^{38}…cycles(3)=[3,9,7,1]=4…remainder:38/4=2…units:2nd=[9]$$
$$units:7^{15}…cycles(7)=[7,9,3,1]=4…remainder:15/4=3…units:3rd=[3]$$
$$units:9^{19}−7^{15}=9-3=6$$

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Re: What is the units digit of 9^19 - 7^15?  [#permalink]

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08 Oct 2019, 04:56
Ans D.. Use cyclicity concept
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Re: What is the units digit of 9^19 - 7^15?  [#permalink]

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08 Oct 2019, 05:11
1
9^1=9
9^2=.1
...
These units digits are repeated in every two terms.
—> If the power of 9 is odd, units digit ends with 9.
—> if the power of 9 is even, units digit ends with 1.
——>Units digit of 9^19 ends with 9

7^1=7
7^2=.9
7^3=..3
7^4=..1
These units digits are repeated in every four terms.
—> units digit of 7^15 is the same as that of 7^3 —> ...3

—> ...9 —3=...6

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Re: What is the units digit of 9^19 - 7^15?  [#permalink]

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08 Oct 2019, 12:33
.....9-.....3=....6
Hence, option D

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Re: What is the units digit of 9^19 - 7^15?  [#permalink]

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08 Oct 2019, 13:49
1
The powers of 9 have the following pattern:
9^1 = 9
9^2 = 81
9^3 = 729
9^4 = 6,561
9^5 = 59,049

As you can see, all even powers will have a units digit of 1 and odd powers will have a units digit of 9.

9^19, 19 is odd so units digit is 9.

Powers of 7 repeat itself after 4 transitions, for example:
7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2,401
7^5 = 16,807

We see that after the fourth power the pattern starts over again, so:
7^15, we take 15 and divide it by four leaving us with 3, the units digit is the 3rd in the repeating list -> 3.

9-3 = 6

IMO
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Re: What is the units digit of 9^19 - 7^15?  [#permalink]

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11 Oct 2019, 15:54
1
Bunuel wrote:

Competition Mode Question

What is the units digit of $$9^{19} - 7^{15}$$?

A. 2
B. 4
C. 5
D. 6
E. 7

The patter of units digits for a base of 9 is:

9^1 = 9

9^2 = 1

9^3 = 9

So, 9 raised to an odd power results in a units digit of 9.

The pattern of units digits for a base of 7 is:

7^1 = 7

7^2 = 9

7^3 = 3

7^4 = 1

7^5 = 7

Thus, we see that 7^4n (where n is an integer) has a units digit of 1. Thus, 7^16 has a units digit of 1, and 7^15 has a units digit of 3. Thus, the units digit of 9^19 - 7^15 is 9 - 3 = 6.

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Re: What is the units digit of 9^19 - 7^15?   [#permalink] 11 Oct 2019, 15:54
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