Saturday, July 4, 2015

POWER OF THE FOURTH!

Using only "mental math" explain why 173^4+179^4+183 is not prime.

Should 5th graders be expected to understand this?

rohitn_wizard said...

We can find the last digit of the result using following method -

(a) Last digit of 173^4 can be calculated by finding the last digit of 3^4 (=81, hence last digit = 1)

(b) Last digit of 179^4 can be calculated by finding the last digit of 9^4 (=81^2, hence last digit = 1)

(c) Adding the last digit of 183, i.e. 3 to the above leads to a number whose last digit is 5.

Any number containing 2 or more digits and ending in 5 cannot be a prime.

Dave Marain said...

Thank you rohitn_wizard. Clear concise explanation. So at what grade level should this be given? 6th? 7th? Of course it isn't just about exponent skill or divisibility, is it? Without having seen models of this kind of thinking, it would be challenging at any level...

rohitn_wizard said...

The students have to understand or know the following aspects
1. How last digit points to trivial divisibility such as last digit being even or 5 or 0.
2. How multiplication or raising to a power affects last digit of a number.
3. What it means for a number to be a prime number or a composite number.
4. Understand how last digit can be calculated for powers without actually finding the exact answer.

More importantly, students need to understand not to be intimidated by large numbers or problems that use large numbers, as there are tricks available to find patterns without tedious calculations.

I am not aware of school syllabus in the US, it seems like a problem for 9th grade.
Some good students in 7th grade may be able to grasp it.

Dave Marain said...

Agreed and nice analysis. My experience is that younger children can be engaged by math "puzzles" particularly involving large numbers. I often them questions like "If you stack a million dollar bills wou askld it reach the ceiling, top of the Empire State bldg, the moon or???" I don't use the word "tricks" as much anymore. I want them to see math as a way of thinking which makes a seemingly hard problem into a manageable one. Are you in the UK?

rohitn_wizard said...

That sounds interesting. No...I am an Indian working in the US.

Dave Marain said...

Thanks for sharing your thoughts and good luck with your work here...

Dave Marain said...

Thanks for sharing your thoughts and good luck with your work here...

rohitn_wizard said...

Thanks. Here's my website - www.rohitn.com - where I have posted explanations of a few formulae/pattern I discovered, along with articles of fractional dimension, etc. Feel free to direct any student who seems inquisitive enough.

One problem in particular (How many times a clock's hour and minute hands meet in a 12 hr period) seems like a common problem. But looking for a pattern to simplify the problem led to a general formulae and an interesting discovery in symmetry. I would encourage students to follow similar thought processes when they encounter any puzzle/problem - finding a general pattern rather than being satisfied by solving the given puzzle/problem.

Here's the link to that article.
http://www.rohitn.com/math/math_clockHandsMeet.aspx

Dave Marain said...

Thank you for sharing. We seem to share many common perspectives. In fact I posted a clock problem a while back and attempted to guide students toward a general formula.