Derek's phone number, $336$ - $7624,$ has the property that the three-digit prefix, $336,$ equals the product of the last four digits, $7 \times 6 \times 2 \times 4.$ how many seven-digit phone numbers beginning with $336$ have this property?

60 possible phone numbers.

Step-by-step explanation:

We want to find how may combinations of 4 numbers a, b, c, d have the product:

a*b*c*d = 336

we know that those numbers can be 7624.

The permutations of those 4 numbers can be:

4*3*2.

We can find other combinations of 4 numbers by the following.

Every number can be written as a product of prime numbers, here, for example, we have that 336 = 7*3*2^4

So we can play with the twos to get different combinations (with the 3 and the 7 we can not play because we have only one of those and we can not divide them into whole numbers)

Other 4 possible numbers are:

7*3*4*4

where i did:

7*6*2*4 = 7*(2*3)*2*4 = 7*3*4*4

where the number of permutations is:

4*3 (because one number repeats itself)

other combination is:

7*6*1*8

where we again have the number of permutations:

4*3*2

The total number of combinations is:

4*3*2 + 4*3*2 + 4*3 = 60

So we have 60 different possible numbers.

40 i don’t know lololol