Which of the following could be the units digit of 57, where n is a positive int

Question

Which of the following could be the units digit of 57, where n is a positive integer? Indicate all such digits. 4. ?] 5 C 2 H 7 [? 8 ?3 ?4 Explanation The units digit of 57" is the same as the units digit of 7 for all positive inte- gers n. To see why this is true forn 2, compute 572 by hand and observe how its units digit results from the units digit of 72, Because this is true for every positive integer n, you need to consider only powers of 7. Beginning with 1 and proceeding consecutively, the units digits of 7, 7, 73, 74 and 7 are 7, 9,3, 1 and 7, respectively. In this sequence, the first digit, 7, appears again, and the pattern of four digits, 7,9,3, 1, repeats without end. Hence, these four digits are the only possible units digits of 7 and therefore of 57 The correct answer consists of Cholces B (1), D (3), H (7), and 3 (9) This explanation ses the following strategies Strategy 7:Find a Pattern Strategy 12: Adapt Solutions to Related Problems Namarie Entrv Duaetions

Explanation / Answer

Here we have 57n

Now n can take values 0,1, 2, 3, 4, 5, 6, 7,....

In all the possible values of n, we are actually looking at what we get when we take powers of 7 as units place of 57n consists of powers of 7

So for n=0, we get 570 = 1

Hence for n=0, units digit is 1 (B)

Now for n=1, 571 = 57

Hence for n=1, units digit is 7 (H)

for n =2, 572 = 57 x 57

So units place will be result of 72 = 49

Hence for n=2, units digit is 9 (J)

for n =3, 573 = 57 x 57 x 57

so units place will be result of 73 = 343

Hence for n=3, units digit is 3 (D)

for n=4, 574 = 57 x 57 x 57 x 57

So units place will be result of 74 = 2401

for n=5, units digit is 1 (B)

for n =5, 575 = 57 x 57 x 57 x 57 x 57

So units place will be result of 75 = 16807

Hence for n=5, units digit is 7 (H)

We can see the pattern is B H J D

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