# ACT Aspire Math: Searcing for Exponential Patterns

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A pattern exists among the units digits of the powers of 3, as shown below. What is the units digit of 3⁹⁵?

ACT Aspire | Math |

ACT Aspire Math | Operations & Algebraic Thinking |

Math | Operations & Algebraic Thinking |

Product Type | ACT Aspire |

Test Prep | ACT Aspire ACT Aspire Math |

### Transcript

The unit's digit of nineteen thousand six hundred eighty three

is three No one wants to figure out three to

the ninety fifth by hand So it's calculator time Yeah

of course This on lee seems like a good idea

until we realized that the scientific calculator on ly presents

answers in scientific notation which doesn't help us find ah

units digit like this thing right here Instead we gotta

look for a pattern in the powers of three which

is a well with problems suggested in the first place

The digits cycle through a one three nine and then

seven before beginning again we see the pattern there So

the task now is to create a formula that tells

us which of these units digits applies to any given

number without counting to ninety five In groups of four

we have three to zero through the fourth and then

through the eighth all of those have units digits of

one notice that the exponents of these terms are all

evenly divisible by four Meanwhile we have three of the

first real fifth and three to the ninth and they

all have units digits of three Well the exponents of

thes terms aren't divisible by four but they all share

a remainder of one when divided by four So that's

the pattern when divided by four exponents with remainder sze

of to have units digits of nine while exponents with

reminders of three have units digits of seven All right

now let's give ninety five a shot Well ninety five

divided by four is twenty three remainder three right So

since the remainder is three the unit's digit is seven

So there we go The answer here is d It's

got to be seven right See have it gets it

gets you cleverly there in number four It's always a 00:01:58.213 --> [endTime] seven That's it That was kind of tricky