heeeeeeeeeeeeeelp maths
posted by Anubhav .
How many integers 1≤N≤1000 can be written both as the sum of 26 consecutive integers and as the sum of 13 consecutive integers?

since they are consecutive integers, the common difference in both series must be 1
d = 1
let the first term of the longer series be a
let the first term of the shorter series be b
sum(n) = (n/2)(2a + (n1)d )
(26/2) (2a + 25(1) ) = (13/2)(2b + 12(1) )
26(2a+25) = 13(2b+12)
52a + 650 = 26b + 156
52a + 494 = 26b
b = (52a + 494)/26 = 2a + 19
if a=1 , b = 21
Long series: 1+2+3+...+26 = 351
short series: 21+22+..+33 = 351
if a=2 , b = 23
Long series: 2+3+..+27 = 377
short series: 23+24+...+35 = 377
.....
the last term in the sort series has to be ≤ 1000 , and clearly odd, making
b = 987 and a = 484
short series: 987 + 988 + ... + 999 = 12909
long series : 484+485+...+509 = 12909
since a goes from 1 to 484 , there are 484 such series.
I will leave it up to you to answer the question. 
Good job reiny
visit on brilliant(dot)org
thank you
Respond to this Question
Similar Questions

quad. eq.
find 3 consecutive integers such that the product of the second and third integer is 20 Take three integers x, y, and z. The for xyz, we want y*z = 20 The factors of 20 are 20*1 10*2 5*4. 20*1 are not consecutive. 10*2 are not consecutive. … 
math
Consecutive integers are integers that follow each other in order (for example 5, 6, and 7). The sum of three consecutive integers is 417.Let n be the first one. Write an equation that will determine the three integers. 
Math
When you solve questions like "The sum of 3 consecutive integers is 147. Find the integers." do you find consecutive even integers and consecutive odd integers the same way? 
math
i don't get this question consecutive integers are integers that differ by one. you can represent consecutive integers as x,x+1,x+2 and so on. write an equation and solve to find 3 consecutive integers whose sum is 33 
math
i don't get this question consecutive integers are integers that differ by one. you can represent consecutive integers as x,x+1,x+2 and so on. write an equation and solve to find 3 consecutive integers whose sum is 33 how do you solve … 
Maths
The number 1000 can be written in several ways as a sum of one or more consecutive positive integers, for instance, 1000=1000 (one summand) or 1000=198+199+200+201+202 (five summands). Find the largest possible number of summands in … 
math plllls heelp
How many integers 1≤N≤1000 can be written both as the sum of 26 consecutive integers and as the sum of 13 consecutive integers? 
math
What is the smallest of three consecutive integers if the sum of the smaller two integers is equal to 177 less than four times the largest? 
math
What is the smallest of three consecutive integers if the sum of the smaller two integers is equal to 177 less than four times the largest? 
math
What is the smallest of three consecutive integers if the sum of the smaller two integers is equal to 177 less than four times the largest?