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

Question

Competition Mode Question

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

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

Sabbir22 · Answer

Cyclic order of 9 raised to the power is

So, the difference of the said expression is 2

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eakabuah · Answer

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.

unraveled · Answer

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

Answer D.

joohwangie · Answer

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

exc4libur · Answer

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

Answer (D)

akshat3010 · Answer

Ans D.. Use cyclicity concept

lacktutor · Answer

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 
The answer is D

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Mohammadmo · Answer

.....9-.....3=....6
Hence, option D

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Kalubalu · Answer

The powers of 9 have the following pattern:
9^1 =  9 
9^2 = 8 1 
9^3 = 72 9 
9^4 = 6,56 1 
9^5 = 59,04 9

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 = 4 9 
7^3 = 34 3 
7^4 = 2,40 1 
7^5 = 16,80 7

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

ScottTargetTestPrep · Answer

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.

Answer: D

Farina · Answer

Cyclicity Concept:
Pattern of 9^2: 9,81,729 --> unit digit pattern is (9,1,9,1..)
pattern is repeated after every second term so 19/2= quotient is 9 and remainder is 1 (R1)

Pattern of 7^2: 7,49, 343, 2401, 16807 --> unit digit pattern is (7,9,3,1,7...)
pattern is repeated after every fourth term so 15/4= quotient is 3 and remainder is 3 (R3)

We will match the remainders with pattern position:
R1's position is 1 in the pattern of 9 i.e ---> at 1 position there exists 9
R3's position is 3 in the pattern of 7 i.e ---> at 3 position there exists 3

so 9 -3 = 6

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

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

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