If m and n are two natural numbers and mn = 32, then find mn+m andnmn.(where n>1)3

1. Break Down the Problem

We need to solve for the expressions $mn + m$ and $nmn$ given that $mn = 32$ and $n > 1$. First, we'll determine the pairs of natural numbers $(m, n)$ that satisfy the equation $mn = 32$.

2. Relevant Concepts

The problem involves finding factors of $32$ to express $m$ and $n$. The natural pairs that multiply to $32$ will help in calculating the required expressions.

3. Analysis and Detail

The pairs of natural numbers $(m, n)$ such that $mn = 32$ are:

• $(1, 32)$
• $(2, 16)$
• $(4, 8)$
• $(8, 4)$
• $(16, 2)$
• $(32, 1)$

Since $n > 1$, we will only consider the pairs:

• $(2, 16)$
• $(4, 8)$
• $(8, 4)$
• $(16, 2)$

Now we can calculate $mn + m$ and $nmn$ for applicable pairs:

1. For $(2, 16)$:

   • $mn + m = 32 + 2 = 34$
   • $nmn = 16 \times 32 = 512$

2. For $(4, 8)$:

   • $mn + m = 32 + 4 = 36$
   • $nmn = 8 \times 32 = 256$

3. For $(8, 4)$:

   • $mn + m = 32 + 8 = 40$
   • $nmn = 4 \times 32 = 128$

4. For $(16, 2)$:

   • $mn + m = 32 + 16 = 48$
   • $nmn = 2 \times 32 = 64$

4. Verify and Summarize

We have calculated both expressions for all relevant pairs ensuring $n > 1$. Now we summarize the results:

• For $(2, 16)$: $mn + m = 34$, $nmn = 512$
• For $(4, 8)$: $mn + m = 36$, $nmn = 256$
• For $(8, 4)$: $mn + m = 40$, $nmn = 128$
• For $(16, 2)$: $mn + m = 48$, $nmn = 64$

Final Answer

The values of $mn + m$ and $nmn$ for each applicable pair are:

• $(2, 16)$: $34, 512$
• $(4, 8)$: $36, 256$
• $(8, 4)$: $40, 128$
• $(16, 2)$: $48, 64$