The number 75 can be written as the sum of the squares

[trapanister]

Indeed this challenging PS is what Ron try to teach in each of his study hall: try, upfront analysing and so on and so forth (by the Thanks Ron I saw all your lessons - 52 recordings - in few days. Amazing).

Coming to this problem: 75 is a multiple of 5 so 5 has to be one of our numbers.

5^2 = 25. this lefts us with 50. From 1 to 9 only 7^2 pemits to have a sum with another number to have a TOTAL of 3 numbers.

5^2=25
7^2=49
From this we need only onother number to cover our "hole" ........1.
1^2=1
Tot = 75.

Indeed 13.

[jnelson0612]

Very nice thinking . . . exactly! :-)

[rihanna.hayat]

if you're interested (and hardcore):
all squares of odds give remainder = 1 upon division by 4. (try it if you don't believe me)
all squares of evens give remainder = 0 upon division by 4.
(if you reallyreallyreally want to know, i can show you why these rules work, although the proofs will be 100% irrelevant to the gmat)
since 75 has remainder = 3 upon division by 4, it follows that all three numbers are odd. as evidenced in the lists above, that reduces your shortlist of candidates to two possibilities.

Thanks a ton Ron. This is an amazing property....I will surely keep in mind while solving any question involving squares. I know trial-error is better method...but anyway i loved this new way of using 'division of 4'.

What I could not understand is my friend's theory:

Coming to this problem: 75 is a multiple of 5 so 5 has to be one of our numbers.

how can be we so sure that square of 5 will be one of the 3 numbers that were added to make 75. Which property am I missing here? Can you please explain.

Thanks.

[tim]

We can't. There is no such property.

[RamonO3]

Thanks Ron. In my case I built a table:
n - n*n
1 - 1
2 - 4
3 - 9
4 -16
5 -25
6-36
7-49
8-64
Then 75-64=11 (there are not option than others 2 n*n sume 11),
so 75-49=26 = 25 + 1, o the numbers are 7,5 and 1.
That took me as 1 minute.

[RonPurewal]

that's a good way to organize the situation.

did you have a question? if so, i can't tell. please clarify if so.

thanks.