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Try to explain it

super_eleman(Super Eleman)

Say when you get a "x", let "k" is the first occurance in which
(7x+k-1) and (k) is NOT in lowest terms
we can say 7x+k-1 = r*p and k=r*q in which r is a prime number
then you will get 7x+r*q-1=r*p and r=(7x-1)/(p-q)
now you get 7x-1 has a factor of r
In your question, x must meet the requirement:
The smallest prime factor of 7x-1 should be bigger than 301. Of course x should be at least 43
and you will easily exclude some cases like x=2n+1,3n+1,5n+3 beause in these cases 7x-1 contains small prime factor 2,3,5
Then the only work you should do is to check
44,50,54,56,60 and you will find 7*54-1=377=13*29 and 7*56-1=391=17*23 which should also be excluded
The remaining numbers are the answer
7*44-1 = 307; 7*50-1 = 349; 7*60-1=419. They are all prime numbers

(#2686511@0)
Last Updated: 2005-12-29
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枫下家园 / 望子成龙 / 征寻数学才子，求解数学难题。不胜感激！ -lilyflower(tiger); 2005-12-29 {680} (#2685883@0)
1.The number of positive integers x with x ≤60 such that each of the rational expressions
7x+1 7x+2 7x+3 7x+300
_____ ______ ______ …, ________
2 3 4 301
is in lowest terms (i.e in each expression, the numerator and denominator have no common factors ) is
A 1 B2 C3 D4 E5

2.In how many ways can a.b.c and d be chosen from the set (0,1,2,...9) so that a <b<c<d and a+b+c+d is multiple of three?

A 54 B64 C63 D90 E72

sorry -lilyflower(tiger); 2005-12-29 {326} (#2685895@0)
1 7x+1 7x+2 7x+3 7x+300
_____, _____, _____, ........, ________
2 3 4 301
I can solve #2. -daceasy(DacEasy); 2005-12-29 {126} (#2685986@0)
You should choose 2 from 0,3,6,9, 1 from 1,4,7, and 1 from 2,5,8, so the unswer should be 6*3*3=54.

I can't understand #1

Update -daceasy(DacEasy); 2005-12-29 {73} (#2686020@0)
The correct unswer should be 55, as 0,3,6,9 is a possible choice as well.

Thank you! -lilyflower(tiger); 2005-12-29 (#2686037@0)

NOT CORRECT! -super_eleman(Super Eleman); 2005-12-29 {232} (#2686036@0)
should plus 3 from {1,4,7} and 1 from {0.3.6.9} that's 1x4=4; and plus 3 from {2,5,8} and 1 from (0,3,6,9} that's another 4; and plus 2 form {1,4,7} and 2 from {2,5,8} that's 3x3=9 and plus 4 from {0,3,6,9} that's 1
TOTAL is 72

I was wrong. -daceasy(DacEasy); 2005-12-29 {41} (#2686043@0)
It shoud be be 54 + 1 + 4 + 4 which is 60

wrong again. Should be 63. -daceasy(DacEasy); 2005-12-29 (#2686050@0)

Thanks -lilyflower(tiger); 2005-12-29 (#2686049@0)
Yes. 72 is correct. -daceasy(DacEasy); 2005-12-29 (#2686059@0)

what about (0,3,6,9)? -goodbaby(小宝); 2005-12-29 (#2686065@0)

Correct question 1 -lilyflower(tiger); 2005-12-29 {291} (#2686072@0)
The number of positive integers x with x ≤60 such that each of the rational expressions
(7x+1)/2, (7x+2)/3, (7x+3)/4, .......... (7x+300)/301
is in lowest terms (i.e in each expression, the numerator and denominator have no common factors ) is
(A )1, (B) 2, (C) 3, (D) 4 or (E) 5.

The answer is 3 -super_eleman(Super Eleman); 2005-12-29 {48} (#2686171@0)
44,50,60. it's too complicate for the deduction

Thank you very much. Can you try to explain why? -lilyflower(tiger); 2005-12-29 (#2686226@0)

Try to explain it -super_eleman(Super Eleman); 2005-12-29 {781} (#2686511@0)
Say when you get a "x", let "k" is the first occurance in which
(7x+k-1) and (k) is NOT in lowest terms
we can say 7x+k-1 = r*p and k=r*q in which r is a prime number
then you will get 7x+r*q-1=r*p and r=(7x-1)/(p-q)
now you get 7x-1 has a factor of r
In your question, x must meet the requirement:
The smallest prime factor of 7x-1 should be bigger than 301. Of course x should be at least 43
and you will easily exclude some cases like x=2n+1,3n+1,5n+3 beause in these cases 7x-1 contains small prime factor 2,3,5
Then the only work you should do is to check
44,50,54,56,60 and you will find 7*54-1=377=13*29 and 7*56-1=391=17*23 which should also be excluded
The remaining numbers are the answer
7*44-1 = 307; 7*50-1 = 349; 7*60-1=419. They are all prime numbers

One more question -lilyflower(tiger); 2005-12-29 {204} (#2686232@0)
In a convex polygon, exactly five of the interior angles are obtuse. The largest possible number of sides for the polygon is
(A) 7, (B) 8, (C) 9, (D) 10 or (E) 11.
Please give the explanations. Thanks.

Try this -super_eleman(Super Eleman); 2005-12-29 {197} (#2686395@0)
say the side number is N
then the summary is (N-2)*180
5 obtuse, N-5 acute
the maximum is 5*180+(N-5)*90 (a limit which can never be reached)
then (N-2)*180<5*180+(N-5)*90
N<9
the answer is 8

You are a genius! Thank you so much! -lilyflower(tiger); 2005-12-29 (#2686407@0)

非常感谢各位的解答，尤其特别致谢数学才子Super Eleman。 -lilyflower(tiger); 2005-12-29 (#2686935@0)

@Calgary

Try to explain it

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