中一培正數學題(快,要交功課)

2009-03-19 4:14 am

1.Find the largeset four-digit number in which all digits are prime numbers.
2.Let (n) be a positive integer. It is known that 20090124 leaves a remainder of 4 when divided by (n). Find the smallest possible value of (n).
3.If the fraction (a/b) <where (a),(b) are positive is 1. Find the smallest possible value of (b).
4.A rectangle with integral side lengths has area 23. Find its perimeter.
5.In the figure, (ABCD) and (AEFG) are squares with side lengths 12 and 16 respectively. (EG) meets (BC) at (M) and (CD) at (N). Find the area of triangle (CMN).
6.In the figure, (IB) and (IC) are the bisectors of <B and <C of triangle (ABC) respectively. If <BIC = 140 degree and <BAC = x degree, find (x).
7.Let (n) be an integer greater that 1. If the sun of digits of (n*n) and (n*n*n) are both a while the sun of digits of (n*n*n*n) is (b), find the smallest possible value of (b).
8.What is the remainder when the 2009-digit number 1000...0001 is divided by 11?
9.In the grids shown, adjacent lines are 1 unit apart. If a maze-like figure of 120 units, how many right angles would have been drawn?
10.In the figure, (D),(E),(F) are points on (AB), (BC) and (CA) of triangle(ABC) respectively such that (BD)=(BE) and (CE)=(CF). If triangle(ADF) is equilateral and triangle(DEF) = x degree,(x).

回答 (1)

Herman Tam

2009-03-19 5:29 am

✔ 最佳答案

Question 1
Is it required that the four digit number should also be a prime number?
If no, then the number should be 7777
if yes, then 7757

Question 2
it means that 20090124=Qn + 4
where Q is the quotient and n is the dividend, Q and n should be integers; n should be larger than 4 (since the remainder is 4)

20090124 - 4=Qn
20090120 =Qn

the smallest possible of n should be 5

Question 3
as (a/b) < 1
then a < b
since a and b are positive integers
the smallest possible value of a is 1
so the smallest possible value of b is 2

Question 4
the rectangle should be 1 x 23
its perimeter = 2 x (1+23) =48

Question 8
1000........1(total 2009 digits) -2 = 9999....9 (total 2008 digits) which is divisible by 11
so 1000......1-2 = 11Q
10000.....1 = 11Q+2
so 2 is the remainder

For questions other than 1 - 4 and 8, you have not provided enough information, like the figure, hence I cannot answer.

參考: ME

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