Use a factoring tree! For each number write the number on top. Then draw two branches below, and write two numbers that multiply to it (not counting one and itself). In the end you will have broken down that numbers into products of prime numbers. In your example, you would get 15=3*5 and 25=5*5.
To get the least common multiple, just take the "most out of each prime number". Since 15 has a 3, take the 3. 15 also has a 5, but 25 has two 5s, so take two 5s. Finally, multiply what you took. 3*5*5=75. That is the least(first) common multiple.
To get the next four common multiples, just multiply 75 by 2, 3, 4, and then 5.
75*2=150
75*3=225
75*4=300
75*5=375
Your first five common multiples are 75, 150, 225, 300, 375.