SOLUTION TO BRAINTEASER NO. 4 DISCLAIMER This solution is but one of many possible methods for solving this problem. Various other solutions may lead to the same answers. This specific solution was provided by the Institute of Mathematical Sciences and Physics of the University of the Philippines Los Baños.
PROBLEM:
Which of the following integers can be expressed as the sum of 1000 consecutive positive integers? a. 1,472,899,500 b. 2,680,349,100 c. 3,579,543,000 d. 4,836,575,700
SOLUTION: The sum of the first 1000 positive integers is
1000 ×1001 2
= 500500. If we add a certain integer k
to each of the 1000 numbers, we will get 500500 + 1000k, which is the sum of k + 1 to k + 1000. Since we are only adding integral multiples of 1000, the last three digits must remain unchanged. This immediately certifies choice a as the only possible answer since its last three digits are 5, 0, and 0 as exactly in that order. Further checking: 500500 + 1000k = 1,472,899,500 Solving for k: 1000k = 1,472,399,000 k = 1,472,399 Thus, it can be inferred from this that 1,472,899,500 is the sum of the integers from 1,472,400 to 1,472,399.
ANSWER:
a. 1,472,899,500
UNIVERSITY OF THE PHILIPPINES CIVIL ENGINEERING SOCIETY Department of Civil Engineering College of Engineering and Agro-industrial Technology University of the Philippines Los Baños
