Quant Boosters  Shashank Prabhu, CAT 100 Percentiler  Set 4

Q28) A sum of money is distributed among certain people. The second person receives Re.1 more than the first, the third Rs. 2 more than the second, the fourth Rs. 3 more than the third, and so on. The first person gets Re.1 and the last person Rs. 67. What is the total number of people?

General term is n(n1)/2+1
first term is n(n1)/2+1=1.... n=1
nth term is n(n1)/2+1=67... n=12
Total 12 terms. Although, it would be far easier to write down the terms and then get the answer.

Q29) Consider ab, a two digit number and it’s square cde, a three digit number. For how many values of ab is (a+b)^2 = c + d + e ? a, b, c, d and e are all positive integers

c+d+e can be one or more among 1, 4, 9, 16, 25 only.
c+d+e=1, a+b=1, (10)^2=100
c+d+e=4, a+b=2, (11)^2=121, (20)^2=400
c+d+e=9, a+b=3, (30)^2=900, (12)^2=144, (21)^2=441
c+d+e=16, a+b=4, (31)^2=961, (13)^2=169, (22)^2=484
c+d+e=25, a+b=5, nothing is possible
Total 9 cases

Q30) ab is a two digit positive number such that ab is divisble by a as well as b. Find sum of all possible values of ab (in numerical value)

10a+b=ka
10a+b=mb
b=(k10)a
10a+ka10a=mak10ma
kmk+10m=0
k(1m)10(1m)=10
(k10)(m1)=10
(15,3) 1 case (15)
(12,6) 4 cases (12,24,36,48)
(11,11) 9 cases (11,22,33...99)
Total 14 cases

@shashank_prabhu a little elaboration on how n(x) = 2^x ?

@shashank_prabhu but doesnt that repeat possible unit digits ?
shouldnt the unit digits be 10 only?

@shashank_prabhu like i can choose any two numbers in c(100,2) ways
out of which all the cases from 01 to 91 [ c (10,2) ] * 4 are valid ?
which gives me 180 / 450

___  ___  ___  ___  ___  ___  ___  ___
There are a total of 8 steps represented by (___) and let the dash ('') represent if he is stopping at the previous step.There are 7 such dashes. Each dash can take a 0 or 1. 0 indicates, he is stopping at the step immediately before it.
As each dash can take a 0 or 1, the number of ways is 128.
And as the maximum steps u can take is 6, the cases where he takes 7 steps at a time  given by (1,0,0,0,0,0,0) and (0,0,0,0,0,0,1) and 8 steps at a time  given by (0,0,0,0,0,0,0) are eliminated.Therefore it is 125.