Johnny once saw plums hanging, like eggs so big and numbered according to the first natural numbers. He is the first to pick the plum with number $2$. After that, Jantje picks the plum each time with the smallest number $n$ that satisfies the following two conditions: $\bullet$ $n$ is greater than all numbers on the already picked plums, $\bullet$ $n$ is not the product of two equal or different numbers on already picked plums. We call the numbers on the picked plums plum numbers. Is $100 000$ a plum number? Justify your answer.