One trick to dissect such questions is to break up the numbers, unfortunately there is no fixed formula within the current syllabus, but you can use some basic understandings to make your life easier.
Lets break 1,000,343 into 1000,000 and 343 and focus on the ending numbers, so it is more manageable.
What is 343, if you try to factorise 343 you'll find it is 7 x 7 x 7. This will be your first clue that the two primes of 1,000,343, ends with a 7 and a 49. From here on you'll have to trial and error. So lets see what primes ends in 7. First 7, 17, 37, 47, 67, 97, 107. You have dramatically cut down the numbers of trial and error and you'll find 1000343 is 107 x 9349

You can use the formula from A Math syllabus :


a³ + b³ = (a + b)(a² - ab + b²)

Notice that 1000343 = 1000000 + 343


1000000 = 100 x 100 x 100 = 100³
343 = 7 x 7 x 7 = 7³

Letting a = 100 and b = 7 (or the other way round, doesn't matter here)


1000343

= 1000000 + 343

= 100³ + 7³

= (100 + 7)(100² - 100(7) + 7²)

= (107)(10000 - 700 + 49)

= (107)(9349)


These are your two prime factors.