3. Represent the number 80 as the sum of two prime numbers. In how many ways can this be done? Let's remind you that one is not considered a prime number.

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
Solution. We organize the enumeration of all prime numbers from 2 to 73 in the form of a table.

| 1 | 7 | 13 | 19 | 25 | 31 | 37 | 43 | 49 | 55 | 61 | 67 | 73 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 2 | 8 | 14 | 20 | 26 | 32 | 38 | 44 | 50 | 56 | 62 | 68 | 74 |
| 3 | 9 | 15 | 21 | 27 | 33 | 39 | 45 | 51 | 57 | 63 | 69 | 75 |
| 4 | 10 | 16 | 22 | 28 | 34 | 40 | 46 | 52 | 58 | 64 | 70 | 76 |


| 5 | 11 | 17 | 23 | 29 | 35 | 41 | 47 | 53 | 59 | 65 | 71 | 77 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 | 78 |

In the table, composite numbers from 4 to 78 are marked in dark color (1 is neither prime nor composite), and there are 21 prime numbers. In the following two tables, pairs of numbers $(a ; b)$ are listed for which $a+b=80$ and $a$ is a prime number.

| a | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| b | 78 | 77 | 75 | 73 | 69 | 67 | 63 | 61 | 57 |


| a | 29 | 31 | 37 | 41 | 47 | 53 | 59 | 71 |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| b | 51 | 49 | 43 | 39 | 33 | 27 | 21 | 9 |  |

In the second row of the table (b), composite numbers are marked in dark color. Thus, the number 80 can be represented as the sum of two prime numbers in four ways: $80=7+73, 80=13+67, 80=19+61, 80=37+43$.

Answer: $80=7+73, 80=13+67, 80=19+61, 80=37+43$; four ways.