Determina il M.C.D. e il m.c.m. dei seguenti monomi con $$ n, m $$ appartenenti a e maggiori di 1 . $$ 356 a^{2} b^{n+2}, \quad a^{n+2} b^{2} $$ $$ 358 a b^{2 n} c^{3}, \quad a^{n} b^{n-1} c^{3 n}, \quad a^{n-1} b^{2 n} c^{4 n} $$ $$ 357 y^{m} z^{2 n}, \quad y^{m-1} z^{2 n+1}, \quad y^{2 m} z^{n} $$ $$ 359 a^{\left(n^{2}\right)} b^{2+m} c^{5 n-5}, \quad a^{n} b^{m-1} c^{n}, \quad a b^{2 \pi} c $$

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Per determinare il massimo comune divisore (M.C.D.) e il minimo comune multiplo (m.c.m.) dei monomi forniti, seguiamo i seguenti passaggi:

### 1. M.C.D. e m.c.m. dei monomi $$ 356 a^{2} b^{n+2}, \quad a^{n+2} b^{2} $$

**M.C.D.:**
- Coefficienti: $$ \text{M.C.D.}(356, 1) = 1 $$
- Parte letterale:
  - $$ a: \min(2, n+2) = 2 $$
  - $$ b: \min(n+2, 2) = 2 $$

Quindi, il M.C.D. è:
$$
\text{M.C.D.} = 1 a^{2} b^{2}
$$

**m.c.m.:**
- Coefficienti: $$ \text{m.c.m.}(356, 1) = 356 $$
- Parte letterale:
  - $$ a: \max(2, n+2) = n+2 $$
  - $$ b: \max(n+2, 2) = n+2 $$

Quindi, il m.c.m. è:
$$
\text{m.c.m.} = 356 a^{n+2} b^{n+2}
$$

### 2. M.C.D. e m.c.m. dei monomi $$ 358 a b^{2 n} c^{3}, \quad a^{n} b^{n-1} c^{3 n}, \quad a^{n-1} b^{2 n} c^{4 n} $$

**M.C.D.:**
- Coefficienti: $$ \text{M.C.D.}(358, 1, 1) = 1 $$
- Parte letterale:
  - $$ a: \min(1, n, n-1) = 1 $$
  - $$ b: \min(2n, n-1, 2n) = n-1 $$
  - $$ c: \min(3, 3n, 4n) = 3 $$

Quindi, il M.C.D. è:
$$
\text{M.C.D.} = 1 a^{1} b^{n-1} c^{3}
$$

**m.c.m.:**
- Coefficienti: $$ \text{m.c.m.}(358, 1, 1) = 358 $$
- Parte letterale:
  - $$ a: \max(1, n, n-1) = n $$
  - $$ b: \max(2n, n-1, 2n) = 2n $$
  - $$ c: \max(3, 3n, 4n) = 4n $$

Quindi, il m.c.m. è:
$$
\text{m.c.m.} = 358 a^{n} b^{2n} c^{4n}
$$

### 3. M.C.D. e m.c.m. dei monomi $$ 357 y^{m} z^{2 n}, \quad y^{m-1} z^{2 n+1}, \quad y^{2 m} z^{n} $$

**M.C.D.:**
- Coefficienti: $$ \text{M.C.D.}(357, 1, 1) = 1 $$
- Parte letterale:
  - $$ y: \min(m, m-1, 2m) = m-1 $$
  - $$ z: \min(2n, 2n+1, n) = n $$

Quindi, il M.C.D. è:
$$
\text{M.C.D.} = 1 y^{m-1} z^{n}
$$

**m.c.m.:**
- Coefficienti: $$ \text{m.c.m.}(357, 1, 1) = 357 $$
- Parte letterale:
  - $$ y: \max(m, m-1, 2m) = 2m $$
  - $$ z: \max(2n, 2n+1, n) = 2n+1 $$

Quindi, il m.c.m. è:
$$
\text{m.c.m.} = 357 y^{2m} z^{2n+1}
$$

### 4. M.C.D. e m.c.m. dei monomi $$ 359 a^{\left(n^{2}\right)} b^{2+m} c^{5 n-5}, \quad a^{n} b^{m-1} c^{n}, \quad a b^{2 \pi} c $$

**M.C.D.:**
- Coefficienti: $$ \text{M.C.D.}(359, 1, 1) = 1 $$
- Parte letterale:
  - $$ a: \min(n^{2}, n, 1) = 1 $$
  - $$ b: \min(2+m, m-1, 2\pi) = m-1 $$
  - $$ c: \min(5n-5, n, 1) = 1 $$

Quindi, il M.C.D. è:
$$
\text{M.C.D.} = 1 a^{1} b^{m-1} c^{1}
$$

**m.c.m.:**
- Coefficienti: $$ \text{m.c.m.}(359, 1, 1) = 359 $$
- Parte letterale:
  - $$ a: \max(n^{2}, n, 1) = n^{2} $$
  - $$ b: \max(2+m, m-1, 2\pi) = 2+m $$
  - $$ c: \max(5n-5, n, 1) = 5n-5 $$

Quindi, il m.c.m. è:
$$
\text{m.c.m.} = 359 a^{n^{2}} b^{2+m} c^{5n-5}
$$

### Risultati finali

1. **M.C.D.**: $$ 1 a^{2} b^{2} $$ e **m.c.m.**: $$ 356 a^{n+2} b^{n+2} $$
2. **M.C.D.**: $$ 1 a^{1} b^{n-1} c^{3} $$ e **m.c.m.**: $$ 358 a^{n} b^{2n} c^{4n} $$
3. **M.C.D.**: $$ 1 y^{m-1} z^{n} $$ e **m.c.m.**: $$ 357 y^{2m} z^{2n+1} $$
4. **M.C.D.**: $$ 
1. **M.C.D.**: $$ a^{2} b^{2} $$ e **m.c.m.**: $$ 356 a^{n+2} b^{n+2} $$ 2. **M.C.D.**: $$ a^{1} b^{n-1} c^{3} $$ e **m.c.m.**: $$ 358 a^{n} b^{2n} c^{4n} $$ 3. **M.C.D.**: $$ y^{m-1} z^{n} $$ e **m.c.m.**: $$ 357 y^{2m} z^{2n+1} $$ 4. **M.C.D.**: $$ a^{1} b^{m-1} c^{1} $$ e **m.c.m.**: $$ 359 a^{n^{2}} b^{2+m} c^{5n-5} $$

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