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well the simplest thing to do would be to go through all the prime numbers and see which ones it's divisible by, again...and again...and again.
You can also write a program/or find a program/or use rules of divisibility to find the largest prime factor.
So, just guessing and checking?
nope-there are rules
if all the digits sum to a multiple of three, the number is divisible by three.
if you take the last digit, double it, and subtract from the remaning digits (and do this over and over) and get a number that's divisible by 7, it has 7 as a factor.
if you sum every other digit, and subtract this from the sum of the OTHER digits, if it's divisible by 11, or is 0, the number has 11 as a factor.
there are tons of rules, and ways you tell a number's factors.
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using a factor tree can be useful - put the number at the top, draw lines down to factors you can recognise, then lines down from those to THEIR factors... until the bottom row of numbers are all primes. The biggest number in that bottom row is what you're after.