Brainsteem Mathematics Challenges - Brilliant Numbers
The Question
A brilliant number is defined as a natural number that is the product of just two primes, both of which have the same number of digits in base 10.
For example, 15 (=3x5) and 35 (5x7) are 2-digit brilliant numbers but 33 (=3x11) is not; 143 (=11x13) is a 3-digit brilliant number.
Find the smallest and the largest 4-digit brilliant numbers.
This is at the level of a lower secondary national mathematics competition.
Try to share your method so as to help each other learn new techniques and strategies.
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Finding the largest 4-digit brilliant number is simple enough. If we multiply a pair of 3-digit numbers together the result will always have at least 5 digits (the lowest possible result being 100 * 100 = 10000). Therefore the highest 4-digit brilliant number is the highest 2-digit prime multiplied by itself. 97 * 97 = 9409.
Finding the smallest 4-digit brilliant number is a bit different. Starting with 1001, we look at the prime factorization of each odd number until we find one that factors into two 2-digit numbers. So the prime factorization of 1001 is 7 * 11 * 13, no good. Next up is 1003, which factors out as 17 * 59. Looks like the answer has been found.
smallest = 1003, largest = 9409
Much faster - very good!
I gave it a try I don't know how to code well to I did it by hand:
First I made a list of the numbers that include all the prime with one decimal point. then with two decimal points which I split up as follows
1)2,3,5,7
2)11,13,17
3)23
4)31, 37
5)41,43,47
I got it by trail and error I got 1147 (31x37)and 2209(47x47). I'm not sure if using 47 twice is possible. Now I want know if its right haha
The two primes can be any 2-digit primes, eg 47x23.
Is the trial and error the only way to solve this or can we use some other technique? I googled brilliant numbers and they had mods. I have little experience with mods though
A lot of these maths competition questions are about reducing the number of options so that enumeration (trial-and-error) can be done quickly.
I will do some modular arithmetic questions soon, but in this case the numbers are quite small.
The way I would start it is to let the two required numbers by pq and xy and rewrite them as
10p + q and 10x + y , so that the digits have become integers. Now multiply them
(10p + q)(10x + y) = ....
If we look for the smallest first, let this equal 1001 and see if we can find pq and xy, if not move on to 1003 and so on. It is actually fairly quick at this point :-)