18. Разложите на простые множители число: 1) 45; 2) 36; 3) 180; 4) 1 200; 5) 1 386.

Решение:

1. 45: \( 45 = 3 × 15 = 3 × 3 × 5 = 3^2 × 5 \).
2. 36: \( 36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3 = 2^2 × 3^2 \).
3. 180: \( 180 = 10 × 18 = (2 × 5) × (2 × 9) = 2 × 5 × 2 × 3 × 3 = 2^2 × 3^2 × 5 \).
4. 1200: \( 1200 = 12 × 100 = (2^2 × 3) × (10^2) = (2^2 × 3) × (2 × 5)^2 = 2^2 × 3 × 2^2 × 5^2 = 2^4 × 3 × 5^2 \).
5. 1386: \( 1386 = 2 × 693 = 2 × 3 × 231 = 2 × 3 × 3 × 77 = 2 × 3^2 × 7 × 11 \).

Ответ: 1) \( 3^2 × 5 \); 2) \( 2^2 × 3^2 \); 3) \( 2^2 × 3^2 × 5 \); 4) \( 2^4 × 3 × 5^2 \); 5) \( 2 × 3^2 × 7 × 11 \).