### Author Topic: Question about 50 lectures book 3  (Read 4301 times)

#### lele_2013 ##### Question about 50 lectures book 3
« on: October 07, 2013, 09:46:59 AM »
Thank you for the solutions to the problems on book 2. Could you please post the solutions to these problems from book 3?

Problem 1. What is the remainder when 5207 is divided by 7?(page 134)
Ans.6

Problem 19. What is the remainder when the sum of the first 102 counting numbers is divided by 5250?
ans.3

Problem 31. What is the remainder when the 2012- digit number 888...888 is divided by
26?
Ans.4

Is there a simpler way to solve this question?
Problem 16. What is the remainder when 10,987,654 is divided by 2,103? (Mathcounts
handbooks)
Ans.1582

Thanks.

#### yongcheng3315

• USAMO Winner
•     • Posts: 2585 ##### Re: Question about 50 lectures book 3
« Reply #1 on: October 07, 2013, 10:23:31 AM »
Thank you for the solutions to the problems on book 2. Could you please post the solutions to these problems from book 3?

Problem 1. What is the remainder when 5207 is divided by 7?(page 134)
Ans.6

Solution:

5207 = 743*7+6. So the remainder is 6.

Problem 19. What is the remainder when the sum of the first 102 counting numbers is divided by 5250?
ans.3

Solution:
1+2+3+...+102 = 5253.
So the remainder is 3.

Problem 31. What is the remainder when the 2012- digit number 888...888 is divided by
26?
Ans.4

Solution:

888888 is divisible by 26.

Since 2012 = 335*6 + 2, the remainder when the 2012- digit number 888...888 is divided by
26 is the the same as the remainder when 88 is divided by 26. Which is 10.

Note: There is a solution to this problem in the book which is correct. 4 is a typo.

Is there a simpler way to solve this question?

Problem 16. What is the remainder when 10,987,654 is divided by 2,103? (Mathcounts
handbooks)
Ans.1582

Solution:
10,987,654 = 5224 *2103 + 1582.
So the remainder is 1582.

Thanks.
« Last Edit: October 07, 2013, 10:24:32 AM by yongcheng3315 »

#### lele_2013 ##### Re: Question about 50 lectures book 3
« Reply #2 on: October 13, 2013, 08:46:27 AM »
Thanks for the solutions. I have some other questions, do you mind posting the solutions?

Problem 6. A triangle has vertices with coordinates A(0, 15), B(0, 0), and C(10, 0). Find
the coordinates of point D on AC so that the area of triangle ABD is equal to the area of
triangle DBC. (Mathcounts Competitions).(page 147 book 3)

Problem 30. The coordinates of one of the endpoints of a diagonal of a rectangle are (–4,2),
and the coordinates of the point of intersection of the diagonals are (1, –1). The sides
of the rectangle are parallel to the axes. What is the number of square units in the area of
the rectangle? (Mathcounts Competitions).
Amswer:60

(1). In the xy- plane, how many lines whose x-intercept is a positive prime number and
whose y-intercept is a positive integer pass through the point (4, 3)? (1994 AMC 12).

Thank you.
« Last Edit: October 13, 2013, 10:27:51 AM by lele_2013 »

#### yongcheng3315

• USAMO Winner
•     • Posts: 2585 ##### Re: Question about 50 lectures book 3
« Reply #3 on: October 13, 2013, 10:01:15 AM »
Thanks for the solutions. I have some other questions, do you mind posting the solutions?

Problem 6. A triangle has vertices with coordinates A(0, 15), B(0, 0), and C(10, 0). Find
the coordinates of point D on AC so that the area of triangle ABD is equal to the area of
triangle DBC. (Mathcounts Competitions).(page 147 book 3)

Solution:

D is the middle point of AC. Using the middle point formula.

Problem 30. The coordinates of one of the endpoints of a diagonal of a rectangle are (–4,2),
and the coordinates of the point of intersection of the diagonals are (1, –1). The sides
of the rectangle are parallel to the axes. What is the number of square units in the area of
the rectangle? (Mathcounts Competitions).
Amswer:60

SOLUTION:

Find the coordinates of the other endpoints of a diagonal of a rectangle using the middle point formula  with two points (–4,2) and (1, –1). Note that  (1, –1) is the  middle point.
Then you can calculate the area.

(1). In the xy- plane, how many lines whose x-intercept is a positive prime number and
whose y-intercept is a positive integer pass through the point (4, 3)? (1994 AMC 12).