Quant Boosters - Sibanand Pattnaik - Set 1
If sum of Factors of N excluding N is equal to N then N is called a perfect number ..
for Eg 28 = 1 + 2 + 4 + 7 + 14
496 = 1 +2 +4 +8 + 16 + 31 + 62 + 124 + 248
PERFECT NUMBERS show another property
The sum of the reciprocal of the factors of a perfect number INCLUDING THE NUMBER ITSELF = 2
Again i repeat "INCLUDING THE NUMBER ITSELF"
for eg : 28 is a perfect number
1+ 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 2
Q28) Find the largest value of a for which 100 + a^3 becomes perfectly divisible by 10 + a
sibanand_pattnaik last edited by zabeer
QA/DILR Mentor | Be Legend
by Factor theorem, if 100 + a^3 is divisible by 10 + a then substitute a = -10 in 100 + a^3 = 100 - 1000 = 900
Now 10 + a must be a factor of 900
so let K * (10 + a) = 900
Now to obtain the greatest value of "a", we have to put K as 1
so 10 + a = 900
=> a = 890
Q29) How many ways can 22 identical balls be distributed in 3 identical boxes ?
It is unordered distribution of a+b+c = 22
so total cases = 24c2
2 same n 1 different cases .. will be .. 0 0 22 ; 1 1 20 ....till 11 11 0 .
so total 0 - 11 that is 12 cases and each of them ll be arranged in 3!/2! i.e 3 ways
why ?? because they are of type A A B ..
so( 24c2 - 36 ) / 6 + 12 = 52
Q30) X and Y have some chocolates with them which they wish to sell . The cost of each chocolate is equal to the number of total chocolates with both of them together initially. Together they sell all the chocolates and after that they start distributing the money collected in this particular fashion. First X takes a 10 rupee note, then Y takes a 10 rupee note and so on. In end it's turn of Y who didn't get any more 10 rupees. How much rupees Y get in his last turn?
As X started the distribution part and took a 10 rupee note first and in the end also, he is able to take a 10 rupee note. That means Total amount, which needs to be a perfect square for n mangoes @ n rupees per mango, is odd multiple of 10 plus some more which is less than 10. That means ten's place digit of the perfect square is ODD. So certainly unit digit of perfect square is 6.
sumit agarwal last edited by zabeer
there is a bit confusion i have.
100 + a^3 = -900 (a = -10)
so how would the answer change here?
zabeer last edited by zabeer
Concept : When a polynomial f(x) is divided by (x - a), remainder is f(a)
Here, f(x) = 100 + x^3
Remainder [f(x)/(10 + x)] = f(-10)
= 100 - 1000 = - 900
So we can write, 100 + x^3 = Q * (10 + x) - 900 ---- (eq 1)
Where Q is some integer (quotient)
Now it is said that f(x) is perfectly divisible by 10 + x.
Means, 100 + x^3 = K * (10 + x) --- (eq 2)
Where K is another integer.
From eq 1 and 2,
Q * (10 + x) - 900 = K * (10 + x)
(Q - K) (10 + x) - 900 = 0
(Q - K)(10 + x) = 900
As we are asked for maximum value of x, we will take (Q - K) as 1.
So (10 + x) = 900
x = 900 - 10 = 890.
Most of the steps here are detailed for the purpose of better understanding and should come intutively otherwise.
sumit agarwal last edited by
number of scores possible = 251 - 15 + 1 ...= 237
Could anyone explain this last step ?