Problem Solving

EDITED

If x,y and z are three consecutive odd numbers and x^2+y^2+z^2=683, what is their sum?

3 numbers 12,13,14 total of their squares is 509 and for 13,14,15 it is 590.

there are no three consecutive numbers with squares adding upto 515.

ok but how to find out that the sum of 12,13,14 squared is 590?

klaud you seem to be lazier than myself
here's the trick 12^2=144 and 13^2= 144+(12+13)
14^2=13^2+(13+14) or 144+(12+13) +(13+14)
this formula works for every consecutive number squared, e.g. 15^2=14^2 +(14+15)
care to learn more short-cuts send me a private message (pm)

klaud wrote:ok but how to find out that the sum of 12,13,14 squared is 590?

klaud wrote:EDITED

If x,y and z are three consecutive odd numbers and x^2+y^2+z^2=683, what is their sum?

Since 20²+20²+20² = 1200, and the required sum here is 683, we know that x, y and z are each less than 20.

The units digit of 683 is 3.
Focus on the units digits of the 3 consecutive odd integers.
Options for the units digits are 1, 3, 5, 7, 9, 1, 3...
The squares of these digits are 1, 9, 25, 49, 81, 1, 9...
Only 9+5+9 = 23 will yield a units digit of 3.
Thus, the 3 integers are 13, 15 and 17.

13+15+17 = 45.

If this question appeared on the GMAT, we could plug in the answers.
When numbers are evenly spaced, median = average.

Answer choice C: 45
Median = 45/3 = 15, implying that the 3 integers are 13, 15 and 17.
13² + 15² + 17² = 169 + 225 + 289 = 683.
Success!

Hi Folks,

Confused a bit !
What about -13, -15 and -17?

Am i missing anything overhere?

Thanks!

garryrother wrote:Hi Folks,

Confused a bit !
What about -13, -15 and -17?

Am i missing anything overhere?

Thanks!

You would hypothetically need to consider negatives in a consecutive integer question if you saw one in the answer choices.

Without picking numbers it can be solved algebraically as well.
Let x be the smallest of the odd numbers <y, so y=x+2 and z>y so z=x+4 {SInce x,y,z are consecutive odd numbers}.
Then,
x^2+(x+2)^2+(x+4)^2=683
or, x^2+x^2+4x+4+x^2+8x+16=683
or, 3x^2+12x=663
or x^2+4x-221=0
or (x+17)(x-13)=0

Hence x is either 13 or -17
So y is either 15 or -15 and z is either 17 or -13.

klaud wrote:EDITED

If x,y and z are three consecutive odd numbers and x^2+y^2+z^2=683, what is their sum?

let me share my solution:

let consecutive odd be n,n+2, n+4
hence
n^2+ (n+2)^2+ (n+4)^2=683
n^2+4n-221=0
hence
n(n+4)=221=17*13
hence n=13, n+2=15, n+4=17

hence 13+15+17= 45

Just a small suggestion to some of the solutions stated above.

We can also use n-2,n,n+2 ... this will help in cancelling out terms and will be marginally quicker.