Here are some references, not a self-contained answer to the question. For self-contained answers, see other people’s answers and comments.

Assuming that symbol ℕ in the question denotes the set of nonnegative integers, your problem is called the feasibility version of the change-making problem. Chapter 5 of Martello and Toth [MT90] states that it is NP-complete, attributing the result to Lueker [Lue75]. I have not checked the report [Lue75].

Even if symbol ℕ in the question denotes the set of positive integers, the problem is still NP-complete because the nonnegative version can be easily reducible to the positive version.

[Lue75] G. S. Lueker. Two NP-complete problems in nonnegative integer programming. Report No. 178, Computer Science Laboratory, Princeton University, 1975.

[MT90] Silvano Martello and Paolo Toth. Knapsack Problems: Algorithms and Computer Implementations, Wiley, 1990. http://www.or.deis.unibo.it/knapsack.html

Here are some references, not a self-contained answer to the question. For self-contained answers, see other people’s answers and comments.

Assuming that symbol ℕ in the question denotes the set of nonnegative integers, your problem is called the feasibility version of the change-making problem. Chapter 5 of Martello and Toth [MT90] states that it is NP-complete, attributing the result to Lueker [Lue75]. I have not checked the report [Lue75].

Even if symbol ℕ in the question denotes the set of positive integers, the problem is still NP-complete because the nonnegative version can be easily reduced to the positive version.

References

[Lue75] G. S. Lueker. Two NP-complete problems in nonnegative integer programming. Report No. 178, Computer Science Laboratory, Princeton University, 1975.

[MT90] Silvano Martello and Paolo Toth. Knapsack Problems: Algorithms and Computer Implementations, Wiley, 1990. http://www.or.deis.unibo.it/knapsack.html

Assuming that symbol ℕ in the question means the set of nonnegative integers, your problem is called the feasibility version of the change-making problem. In Chapter 5 of Martello and Toth [MT90], its NP-completeness is attributed to Lueker [Lue75]. I have not checked the report [Lue75].

(Even if symbol ℕ in the question means the set of positive integers, showing the NP-completeness is not difficult: just reduce the nonnegative version to the positive version.)

[Lue75] G. S. Lueker. Two NP-complete problems in nonnegative integer programming. Report No. 178, Computer Science Laboratory, Princeton University, 1975.

[MT90] Silvano Martello and Paolo Toth. Knapsack Problems: Algorithms and Computer Implementations, Wiley, 1990. http://www.or.deis.unibo.it/knapsack.html

Here are some references, not a self-contained answer to the question. For self-contained answers, see other people’s answers and comments.

Assuming that symbol ℕ in the question denotes the set of nonnegative integers, your problem is called the feasibility version of the change-making problem. Chapter 5 of Martello and Toth [MT90] states that it is NP-complete, attributing the result to Lueker [Lue75]. I have not checked the report [Lue75].

Even if symbol ℕ in the question denotes the set of positive integers, the problem is still NP-complete because the nonnegative version can be easily reducible to the positive version.

[Lue75] G. S. Lueker. Two NP-complete problems in nonnegative integer programming. Report No. 178, Computer Science Laboratory, Princeton University, 1975.

[MT90] Silvano Martello and Paolo Toth. Knapsack Problems: Algorithms and Computer Implementations, Wiley, 1990. http://www.or.deis.unibo.it/knapsack.html