I have to prove that there is no

sempteim245

sempteim245

Answered question

2022-04-01

I have to prove that there is no natural number whose multiplication of digits is equal to 3570
What would be the proper mathematical solution to this question?

Answer & Explanation

Kingston Lowery

Kingston Lowery

Beginner2022-04-02Added 10 answers

Step 1
Suppose there were a natural number that satisfies this. Any natural number has the base 10 digit form x1x2xn where we understand each xj to be a digit (not a product!). In particular 1x19 and is an integer, and for 2jn, 0xj9 and is also an integer. Then we would have
j=1nxj=3570=2×3×5×7×17.
17 is a ' and hence on the left hand side of the above one would need an xj to be a multiple of 17 for the equation to hold. But every multiple of 17 is two digits or more which contradicts the definition of our xjs. Hence we have the result.

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