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If n is a positive integer, and its greatest divisor different from n 70-447 itself equals to 91, casinos how many possible values are there for n?

(A) 3

(B) 4

(C) 37

(D) 90

(E) Infinitely many

If, D=1, N-1=91, N=92 – is not suitable, because 92/1 – no greatest divisor

Roman, could you please add some details to your comment? =)

By the way, here we do not deal with the greatest COMMON divisor, just the greatest divisor of a number.

N – positive integer

D – greatest divisor

So,

N*D-D=91, hence

(1) D(N-1)=13*7 or

(2) D(N-1)=91*1

->(1) D=7, N=14 (N-1=13)

D=13, N=8 (N-1=7) – is not suitable, 1 – is not greatest divisor

->(2) D=1, N=92 (N-1=91) – is not suitable, 1 – is not greatest divisor

D=91, N=0 (N-1=1) – is not suitable, 0 – is not positive integer

Answer: N=14 D=7

Only ONE solution, another thought?

The problem tells us, that 91 is a divisor of n. Doesn’t that contradict the solution above?

Why 91 is a divisor of n?

N*D-D=91

D(N-1)=91 => D=91/(N-1)

Maybe I misunderstood the problem?

It seems, that we are solving different problems indeed. The 1st sentence reads, that n’s greatest divisor different from n itself equals to 91. So, 91 is the divisor of n.

And how do you explain, that N*D-D=91?

I thought, that the difference between number and greatest divisor is equal 91.

If N- positive integer

D = 91 – is the greatest divisor of N (D<N)

Hence,

N=91*prime number – from 2 to 91 (2.3.5.7.11.13.17.19.23.29.31.37.41.43.47.53.59.61.67.71.73.79.83.89) – 24 possible values for N

Because if we use 91*4 we can provide the same as 91*2*2=182*2… so greatest divisor will be 182

But no answers. – 24.

Of all the calculations I tend to only answer D.

confused …

Roman, you almost got it right!

What is the greatest divisor of N=91*19, for example?

N = 91 * 91 = 1729

I think that if we choose two numbers 91 and 19, then 91 – the greatest ,

But, if we divide into prime factors and combine = 91*19 = (7*13)*19 = 7* (13*19) = 7*247=1729 – not suitable, 247>91

So,

1) 91*2=(7*13)*2 – ok, the greatest 91

2) 91*3=(7*13)*3 – ok, the greatest 91

3) 91*5=(7*13)*5 – ok, the greatest 91

4) 91*7=(7*13)*7=7*(13*7) – ok, the greatest 91

5) 91*11=(7*13)*11=7*(13*11)=143 – not suitable, 143>91, hence all prime numbers >7 not suitable.

Correct answer В=4?