+0  
 
+1
25
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Consider two positive even integers less than 15 (not necessarily distinct). When the sum of these two numbers is added to their product, how many different possible values may result?

 
 Jan 12, 2019
 #1
avatar+419 
+3

so it is looking for ab+a+b, if you add one to that, they will still be the same, distinct or not distinct, so it wont change it so you can just make it ab+a+b+1, which factors as (a+1)(b+1). now you are just trying to find those values.

 

a,b=2,4,6,8,10,12,14

i found no way to do this other than casework..

if a = 2 then it equals 9,15,21,27,33,39, and 45

if a = 4 then it equals 15,25,35,45,55,65, and 75

if a = 6 then it equals 21,35,49,63,77,91, and 105

if a = 8 then it equals 27,45,63,81,99,117, and 135

if a = 10 then it equals 33,55,77,99,121,143, and 165

if a = 12 then it equals 39,65,91,117,143,169, and 195

if a = 14 then it equals 45,75,105,135,165,195, and 225

 

counting them, and listing, you get the numbers 9, 15, 21, 27, 33, 39, 45, 25, 35, 55, 65, 75, 49, 63, 77, 91, 105, 81, 99, 117, 135, 121, 143, 165, 169, 195, and 225, for a total of 27 numbers. 

 

HOPE THIS HELPED!!

 

(may be 28, im not sure)

 
 Jan 12, 2019
 #2
avatar+94526 
+1

asdf is correct....

 

Since the integers do not have to be distinct (different).....we have....

 

2*2 + 2 + 2  =  8         4*4 + 4 + 4 =  24        6*6 + 6 + 6   =  48           8*8 + 8 + 8  =    80

2*4 + 2 + 4 =  14        4*6 + 4 + 6  = 34        6*8  +6 + 8  =   62           8*10 + 8 + 10 = 98 

........20                                    ........ 44                           ......76                             ......116

........26                                    .........54                           ......90                             ......134

........32                                    .........64                         ......104

........38                                    .........74

........44

 

 

10*10 + 10 + 10  =   120          12 * 12 + 12 + 12   = 168            14 * 14 + 14 + 14  = 224

10*12 + 10 +  12 =   142           12*14 +  12 + 14   = 194

                           ..... 164

 

We have     (7)*(8) / 2  - 1  =   27 different values.....smiley

 

 

 

cool cool cool

 
 Jan 12, 2019

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