Example 15. Use both the Intersection of set and Prime
Factorization methods
to find GCF (1421, 1827, 2523):
Intersection of set method: GCF (largest factor which divides both)
Prime Factorization method: GCF (lowest prime common to all)
1421 => 72 x 29
1827 => 32 x 7 x 29
2523 => 3 x 292 Common prime factor to all is 29
Example 16. Use the difference theorem to find:
(a) GCF (1847, 1421) (b) GCF(2523, 1827)
GCF(1847, 1427) = GCF(1827-1421, 1421) = GCF(1421, 406) = GCF(1015, 406)
GCF(1015, 406) = GCF(609, 406) = GCF(203, 203) = 203
GCF(1847, 1427) = 203
GCF(2523, 1827) = GCF(2523-1847, 1827) = GCF(1827, 696) = GCF(1131, 696)
GCF(1131, 696)=GCF(696, 435) = GCF(435, 261) = GCF(261, 174) = GCF(174, 87)
GCF(87, 87) = 87
GCF(1847, 1427) = 87
Example 17. Use the reminder theorem to find:
(a) GCF (2523, 1847)
GCF(2523, 1847) => 2523 / 1847 = 1 R 696
1847 / 696 = 2 R 435
696 / 435 = 1 R 261
435 / 261 = 1 R 174
261 / 174 = 1 R 87
174 / 87 = 2 R 0
GCF(2523, 1847) = 87
Example 18. Use both the Intersection of set and Prime Factorization methods to find LCM (15, 35, 42, 80):
Intersection of set method: LCM (smallest multiple of all)
Where A = {15, 30, 45, ....,1680, 1695, ....}
B = {35, 70, 105, ...,1680, 1715, ...}
C = {42, 84, 126, .... 1680, 1722, ..}
D = {80, 160, 240, .. 1680, 1760, ..}
Prime Factorization method: LCM (highest exponent of primes in set)
15 => 3 x 5
35 => 5 x 7
42 => 2 x 3 x 7
80 => 24 x 5
LCM = 15 => 24 x 3 x 5 x 7 = 1680
Example 19. If GCF(2523, 1827) = 87, Find LCM (2523, 1827).
(Theorem: GCF(a,b) x GCF(a, b) = ab)
So: GCF(2523, 1827) x LCM(2523, 1827) = 2523 x 1827 = 4609521
LCM (2523, 1827) = 4609521 / 87 = 52983