How can we be sure that in denominator will always be cancelled by some of the values of numerator.The 3 part is disturbing me. plz help:drinking:
deepak_pgi Says
The same thing is bothering me also. While finding the values, we didn't bother about the 3 in denominator. :lookround:
As per you, for odd values of M other than multiples of 3,(2M+1) cancels the 3. But this is not the case with 23 as 2*23+1=47 , which is not divisible by 3. (M+1=24) will be responsible for the cancellation of 3 in this case.
PS:- Just wanted to bring this case to your notice.
Actually, we know that k = 1 + 2 + ... M is always an ineger, hence we can say that M(M + 1)(2M + 1)/6 will always be an integer.
=> M(M + 1)(2M + 1) wil always be a multiple of 3 whatever be the value of M
So, we just need to find those values of M for wich M(M + 1)(2M + 1) is divisible by 8.
=> M can be 7, 8, 15, 16, 23, 24, 31, 32, 39, 40, 47, 48, 55 => Total 13 values
=> M(M + 1)(2M + 1) wil always be a multiple of 3 whatever be the value of M
As far as above stmt. is considered I got it by pen and paper (by trying combinations for m=1,2,3,4...) How did u realise? Is there any way or theorems etc. Plz elaborate it.
As far as above stmt. is considered I got it by pen and paper (by trying combinations for m=1,2,3,4...) How did u realise? Is there any way or theorems etc. Plz elaborate it.
See, 1 + 2 + ... + M = M(M + 1)(2M + 1)/6
We know that LHS is always an integer, hence RHS will always be an integer.
=> M(M + 1)(2M + 1)/6 will always be an integer, which is only possible if M(M + 1)(2M + 1) will be a multiple of 6.
M is a two digit number which has the property that... the product of factorials of its digits > sum of factorials of its digits.. how many value of M exist
M is a two digit number which has the property that... the product of factorials of its digits > sum of factorials of its digits.. how many value of M exist
There are following possible values that negate this condition: 10,20,30,40,50,60,70,80,90 - 9 nos 11 to 19 - 9 nos 21,31,41,51,61,71,81,91 = 8 nos 22
So out of 90 two digit no 25 dont follow the above condition so answer in 65
The following is a question from the Number Systems chapter - I know the answer given at the back of the book is incorrect - Just wanted to reconfirm:
A boy has a certain number of chocolates. He eats 1/3rd of them and 1 more in the morning and then 1/5th of the remaining and 4 more in the evening after which he is not left with any chocolates. How many chocolates did he have at the beginning
The following is a question from the Number Systems chapter - I know the answer given at the back of the book is incorrect - Just wanted to reconfirm:
A boy has a certain number of chocolates. He eats 1/3rd of them and 1 more in the morning and then 1/5th of the remaining and 4 more in the evening after which he is not left with any chocolates. How many chocolates did he have at the beginning
The following is a question from the Number Systems chapter - I know the answer given at the back of the book is incorrect - Just wanted to reconfirm:
A boy has a certain number of chocolates. He eats 1/3rd of them and 1 more in the morning and then 1/5th of the remaining and 4 more in the evening after which he is not left with any chocolates. How many chocolates did he have at the beginning
a) 12 b) 13 c) 15 d) none of the above.
Here, try with a multiple of 3 and proceed with the question.. If you get a fraction, go for the next multiple.. In this case it is 9
@ achilles first try numbers like 10,2,11 then 10,3,11 like this upto 10,9,11 repeat same process with no 9 then 8 ,7,6,5,4,3,2,1 remember dont count the triangles with same dimensions twice u"l get 20
question from ratio and proportion Brass is an alloy of copper and zinc. bronze is an alloy containing 80% of copper,4% zinc and 16% of tin. A fused mass of brass and bronze is found to contain 74% copper,16% zinc and 10% tin. the ratio of copper to zinc in brass is??? plz reply with complete solution thnks
question from ratio and proportion Brass is an alloy of copper and zinc. bronze is an alloy containing 80% of copper,4% zinc and 16% of tin. A fused mass of brass and bronze is found to contain 74% copper,16% zinc and 10% tin. the ratio of copper to zinc in brass is??? plz reply with complete solution thnks
Take 1 Kg of Bronze it will contain --> 160gms of tin now this 160 gms is 10% of fused mass so total weight of fused mass is 1600gms
Now this 1600 gms contain 74% copper which is 1184 gms subtract 800 from it we get 384 ( 800 of 1000 of bronze)
so from brass which is 0f 600 gms in alloy we have 384 gms of copper and 216 gms of zinc so the ratio is 64:36
A number xy is multiplied by another number ab and the result comes as pqr ,where r=2y, q=2(x+y) and p=2r where p=2x where x,y pls give detailed solution
A number xy is multiplied by another number ab and the result comes as pqr ,where r=2y, q=2(x+y) and p=2r where p=2x where x,y pls give detailed solution
