Questions that were on the PLAN test. (Math)

[ChrisŤ]

Like I said in the topic, these two questions were on the PLAN test a while back (PLAN is a pre-ACT test prep thingy).

They are both math. I asked my brother - who is an Aeronautical engineer, and my Algebra 2 teacher. And neither of them could figure it out.

I would like to know how to do them just in case I see them again.

Here goes...

Let a and b be positive integers. Which of the following describes the product of (3.2 x 10^a)  and (5.3 x 10^b)?
A.  Less than 10^a+b
B. Between 10^a+b and 10^a+b+1
C. Between 10^a+b+1 and 10^a+b+2
D. Between 100^a+b and 100^a+b+1
E. More than 100^ab


*Next question*


The first positive integer powers of 3 are shown below. What is the units digit for 3^215?
3^1= 3
3^2= 9
3^3= 27
3^4= 81
3^5= 243
3^6= 729
3^7= 2187
3^8= 6561




A. 1
B. 3
C. 5
D. 7
E. 9

[jim360]

For the first one, estimate. It's rougly 3x5x10^{a + b} or 1.5x 10^a+b+1, so the answer is C.

For the second, notice that every four times the nubmer at the end of the power of three repeats in order 3, 9, 7, 1. What you should notice is that the last digit is 3 if the power raised by is n=1 (mod 4), and it happens 215 = 3 (mod 4), so the answer is 7, D.

To explain what I just did, n=1 (mod 4) means that "when n is divided by 4, the remainder is 1" (The = should have 3 lines rather than two, since it's not possible to solve for n - this is a congruence, not an equation). So you divide 215 by 4 and find that there is a remainder of 3. (If you need help with this, PM me and I'll try to explain further.)

[Miser]

I got the same answers as jim, for the 1st one, i used the rule of exponents to expand 10^a+b+1 to 10^(a+b) x 10^1  and the same for 10^a+b+2. I found that 3.2X5.3 is in between 10^1 and 10^2.

as for the second i used the same method as jim. 

[ChrisŤ]

Thank you for the quick responses, I sorta get them now, Im sure I will in a little bit.

(My IQ score is lower because I am a slow thinker/slow processing, so the gifted teacher says when I asked why im not in gifted lol)


So the units digit for 3^214 is 9?

[jim360]

yes, that's right.

Any more questions, feel free to PM me.

[BFM_Edison]

I concur with the aforementioned answers

[jim360]

Thanks.

You spammer....

[BFM_Edison]

Was just reinforcing your answers' validity

[Manticore]

My brain hurts... what grade level is this? *makes stupid face*

And yes. thats what all teachers say when you ask why your not in gifted (and talented)...

TOO MUCH INFO: I failed the gifted test twice = I r big retard. 

[BFM_Edison]

Seems to me like I could have answered it when I was taking Algebra 1 (ninth grade math as sixth grader, though).