If 770 is written as the product of three positive integers,

Question

If 770 is written as the product of three positive integers, how many possibilities are there for the largest integer?

(A) 11
(B) 12
(C) 13
(D) 14
(E) 15

paragw · Answer

Here's my solution for this question

kevincan · Answer

Unfortunately, this solution is flawed : 10 cannot be the largest integer , but 11 can be.

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=89461 This is how I solved it - I had to quickly do manual verification rather than apply some technique (nothing much came to mind!). Great question! --- Harsha GMAT-Club-Forum-p17fjc0s.png

kevincan · Answer

For some reason , I can’t see Harsha’s splendid solution any more , but here’s a summary:

770 = 2*5*7*11 has  2^4  factors, which can be listed as 8 factor pairs : 1*770, 2*385 etc. The largest integer in the product of three positive integers can be any of the greater numbers in these 8 factor pairs, as well as three of the smaller numbers : 11,14 and 22.

AngryHook · Answer

Alternative approach:
We can get the total factors being 16, using the prime factor method.
Now, consider the factors: 1, 2,5,7- They surely can’t be the largest. Consider factors above or equal to 10; we get that 10 require an 11 and 7 to give the answer so surely it is eliminated. Similarly, 11 qualifies. Thus outta 16 we eliminated 5, giving 11 as an answer

AngryHook · Answer

Alternative approach:
We can get the total factors being 16, using the prime factor method.
Now, consider the factors: 1, 2,5,7- They surely can’t be the largest. Consider factors above or equal to 10; we get that 10 require an 11 and 7 to give the answer so surely it is eliminated. Similarly, 11 qualifies. Thus outta 16 we eliminated 5, giving 11 as an answer

AngryHook · Answer

Alternative approach:
We can get the total factors being 16, using the prime factor method.
Now, consider the factors: 1, 2,5,7- They surely can’t be the largest. Consider factors above or equal to 10; we get that 10 require an 11 and 7 to give the answer so surely it is eliminated. Similarly, 11 qualifies. Thus outta 16 we eliminated 5, giving 11 as an answer

chetan2u · Answer

What would the smallest value of such number be?
 1. It will be equal to or greater than cube root of the number 770. 
 2. Incase any prime factor is bigger than the above value.

Now  9^3  is 729 and  10^3  is 1000. So, the value has to be greater than 9.
However, the largest prime factor is 11, making 11 the smallest value we are looking for.

Knowing 11 as the least value, let us look for other values. 770 can be written as 2*5*7*11. 
A. Taking one integer out of the four, 2, 5, 7 and 11.
Only possibility 11, that is  1  value.
 B. Taking two integers out of the four, 2, 5, 7 and 11.
4C2 or 6. But 2*5 will be less than 11. So  5  values.
3. Taking three integers out of the four, 2, 5, 7 and 11.
 4C3 or  4  values
D. Taking all four integers out of the four, 2, 5, 7 and 11.
4C4 or  1  value.

Thus total becomes 1+5+4+1 = 11

A

If 770 is written as the product of three positive integers,

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If 770 is written as the product of three positive integers,

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