The seventh grade Mathematical Olympiad Volume I (Seeking Process, Question and Answer)

1. Let A and B be natural numbers and satisfy the relation (1111+A) * (11/)

Prove that a-b is a multiple of 4.

2. Two postmen, A and B, walk in opposite directions at a constant speed. When they met, A walked 18 kilometers more than B. After meeting, A walked 4.5 hours to reach B, and B walked 8 hours to reach A, so as to find the distance between A and B..

1、

First, we observe the result of 123456789. We know that this is an odd number. If we want the product of two numbers to be odd, then both numbers must be odd.

(1111+a) and (111-b) are both odd numbers-conclusion.

Therefore, we can continue to deduce that A and B are even numbers-conclusion (2).

We transform the equation appropriately, as follows:

( 1 1 1 1 1+a)*( 1 1 1 1-b)= 123456789

[( 1 1 1 1 1+b)+(a-b)]*( 1 1 1 1 1-b)= 123456789

( 1 1 1 1 1+b)*( 1 1 1 1-b)+(a-b)*( 1 1 1 1 1-b)= 655438

(a-b)*( 1 1 1 1 1-b)= 2428+b * b

B is an even number, so b*b is a multiple of 4, and 2428 is also a multiple of 4 = = >

(2428+b*b) is a multiple of 4,

And because (1111-b) is an odd number = = > (a-b) is a multiple of 4.

2. Let the distance from the meeting point b be x kilometers, then the speed ratio of A and B = (x+18): x.

The remaining distance ratio between Party A and Party B =x:(x+ 18)