From http://www.sgforums.com/forums/2297/topics/343862
1)
a) Express 560 as a product of its prime factors(in index notation)
b) State the smallest integer value of x for which 560x is a square number
2) Write down the largest factor of 1155 other 1155 itself
3) Find the least integer value of n if 936n is a perfect square
4) A three-digit number is the product of 4 prime number. The sum of its prime factors is 30. Given that the three digits of the number are all prime and different, find the three-digit number.
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Answer:
1)
i) 560= 24 x 5 x 7
ii) We need 5 x 7
Thus, x = 35
2) 1155 = 3 x 5 x 7 x 11
largest factor = 5 x 7 x 11 = 385
3) 936 = 23 x 32 x 13
We need 2 x 13
n = 26
4) This is a brute force method...
Prime numbers are 2, 3 , 5, 7, 11, 13, 17, 19 and so on...
Since the 3 digits must all be prime, they can only be 2, 3, 5, 7. Since we only have 4 possible digits without repetition, there are only 24 combos (4P3= 24).
1. 235 = 5 x 47 (NA) 2 primes
2. 237 = 3 x 79 (NA) 2 primes
3. 253 = 11 x 23 (NA) 2 primes
4. 257 (NA) 1 prime
5. 273 = 3 x 7 x 13 (NA) 3 primes
6. 275 = 52 x 11 (NA) 3 primes
7. 325 = 52 x 13 (NA) 3 primes
8. 327 = 3 x 109 (NA) 2 primes
9. 352 = 25 x 11 (NA) 3 primes
10. 357 = 3 x 7 x 17 (NA) 3 primes
11. 372 = 22 x 3 x 31 (NA) sum not 30
12. 375 = 3 x 53 (NA) sum not 30
13. 523 (NA) 1 prime
14. 527 = 17 x 31 (NA) 2 primes
15. 532 = 22 x 7 x 19 (NA) This is the one
16. 537 = 3 x 179 (NA) 2 primes
17. 572 = 22 x 11 x 13 (NA) sum not 30
18. 573 = 3 x 191 (NA) 2 primes
19. 723 = 3 x 241 (NA) 2 primes
20. 725 = 52 x 29 (NA) 3 primes
21. 732 = 22 x 3 x 61 (NA) sum not 30
22. 735 = 3 x 5 x 72 (NA) sum not 30
23. 752 = 24 x 47 (NA) 5 primes
24. 753 = 3 x 251 (NA) 2 primes
