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A trillion is 10^12. Which of the following is bigger: the three trillionth root of 3 or the two trillionth root of 2? You may assume that if 0 < x < y, then 0 < x^n < y^n for integer values of n greater than or equal to 1.

For this sort of question, the easiest approach is to make an educated guess at the correct answer, and prove that this must be correct by contradiction. As taking the root of something has a large effect on...
BC
Answered by Benjamin C. • MAT tutor
1852 Views

Find the number of solutions x in [0,2pi) to the equation 7sin x +2(cos x)^2 =5.

The problem seems to be hard as the equation involved both cosx and sinx But we can relate the two as (cosx)^2 + (sinx)^2 = 1, so we can rearrange this as 0 = 2(sinx)^2 - 7sinx + 3. This quadratic in sinx ha...
JR
Answered by James R. • MAT tutor
3247 Views

How many 0's are at the end of 100! (100 factorial)?

The number of 0's at the end of a number is the same as the number of factors of 10 that the number has. So consider the prime factor decomposition of 100! and note that every factor of 10 has prime factors ...
HR
Answered by Henry R. • MAT tutor
5023 Views

I've been doing specimen MAT admission test - but I couldn't figure out the answer to the parts III, and IV of question 6 (https://www.maths.ox.ac.uk/system/files/attachments/speca.pdf). Is there some kind of a trick?

As for part III you only need to use the process of elimination. Notice, that there are only three possible situations, since at east one of Alice, Bob, and Charlie has a white hat; namely: one of them has a...
SN
Answered by Stanisław N. • MAT tutor
2817 Views

Let r and s be integers. Then ( 6^(r+s) x 12^(r-s) ) / ( 8^(r) x 9^(r+2s) ) is an integer when: (a) r+s <= 0, (b) s <= 0, (c) r <= 0, (d) r >= s.

First of all, an integer is a whole number, i.e. a not a fractional number, that can be positive, negative or zero. To find the range of values r and s for which the above fraction is an integer we can start...
NP
Answered by Nadia P. • MAT tutor
15958 Views