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the next step is to choose all possibilities for which n>77:

one prime number: 2, 4, 5, 25, 7- 0 possibilities for n two prime numbers: 2*5, 2*25, 4*5, 4*25, 2*7, 4*7, 5*7, 25*7 - 2 possibilities for n three prime numbers: 2*5*7, 4*5*7, 2*25*7, 4*25*7- 3 possibilities for n

So, we have 5 possible values of n: 4*25, 25*7, 4*5*7, 2*25*7, 4*25*7 _________________

the next step is to choose all possibilities for which n>77:

one prime number: 2, 4, 5, 25, 7- 0 possibilities for n two prime numbers: 2*5, 2*25, 4*5, 4*25, 2*7, 4*7, 5*7, 25*7 - 2 possibilities for n three prime numbers: 2*5*7, 4*5*7, 2*25*7, 4*25*7- 3 possibilities for n

So, we have 5 possible values of n: 4*25, 25*7, 4*5*7, 2*25*7, 4*25*7

I meant to post the actual explanation that came with the question.. and which is still confusing to me.. If anybody can evaluate the latter, it will be great (bolded part..):

If the remainder is 77, then n must logically be greater than 77. Also, there must be a positive integer q such that 777= nq + 77. i.e. nq = 700.

Therefore, the factors of 700 greater than 77 comprise the possible values of n. Instead of counting the factors of 700 that are greater than 77, let’s count the ones that are less than or equal to 700/77. As 700 = 50 × 2 × 7, we can see that there are 5 factors of 700 that are less than or equal to 7: 1 , 2 , 4 , 5 , and 7. Thus there are 5 possible values of n (i.e. factors of 700) greater than 77. We don’t need to know these values, but for the curious, they are 700, 350, 175, 140 and 100.

Instead of counting the factors of 700 that are greater than 77, let’s count the ones that are less than or equal to 700/77. As 700 = 50 × 2 × 7, we can see that there are 5 factors of 700 that are less than or equal to 7: 1 , 2 , 4 , 5 , and 7. Thus there are 5 possible values of n (i.e. factors of 700) greater than 77. We don’t need to know these values, but for the curious, they are 700, 350, 175, 140 and 100.

Excellent!

700=(factor >77)*(factor <700/77)

OE say that for each factor >70 we have one factor <700/77 in order to have 700 as a product of those factors. So, we can count only all factors <700/77

Instead of counting the factors of 700 that are greater than 77, let’s count the ones that are less than or equal to 700/77. As 700 = 50 × 2 × 7, we can see that there are 5 factors of 700 that are less than or equal to 7: 1 , 2 , 4 , 5 , and 7. Thus there are 5 possible values of n (i.e. factors of 700) greater than 77. We don’t need to know these values, but for the curious, they are 700, 350, 175, 140 and 100.

Excellent!

700=(factor >77)*(factor <700/77)

OE say that for each factor >70 we have one factor <700/77 in order to have 700 as a product of those factors. So, we can count only all factors <700/77