EDITED
If x,y and z are three consecutive odd numbers and x^2+y^2+z^2=683, what is their sum?
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consecutive numbers
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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.
there are no three consecutive numbers with squares adding upto 515.

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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)
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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 shortcuts send me a private message (pm)
klaud wrote:ok but how to find out that the sum of 12,13,14 squared is 590?
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Since 20Â²+20Â²+20Â² = 1200, and the required sum here is 683, we know that x, y and z are each less than 20.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?
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.
13Â² + 15Â² + 17Â² = 169 + 225 + 289 = 683.
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You would hypothetically need to consider negatives in a consecutive integer question if you saw one in the answer choices.garryrother wrote:Hi Folks,
Confused a bit !
What about 13, 15 and 17?
Am i missing anything overhere?
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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+4x221=0
or (x+17)(x13)=0
Hence x is either 13 or 17
So y is either 15 or 15 and z is either 17 or 13.
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+4x221=0
or (x+17)(x13)=0
Hence x is either 13 or 17
So y is either 15 or 15 and z is either 17 or 13.

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let me share my solution: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 consecutive odd be n,n+2, n+4
hence
n^2+ (n+2)^2+ (n+4)^2=683
n^2+4n221=0
hence
n(n+4)=221=17*13
hence n=13, n+2=15, n+4=17
hence 13+15+17= 45