Question

The number of ordered pairs (P,Q) for which 213P43Q6 is divisible by 99 is

Answer 1

remember divisibility rule for \(9\) ?

Answer 2

sum of all the digits should be a multiple of 9

Answer 3

yup!
setup a equation

Answer 4

\(2+1+3+P+4+3+Q+6 = 9k \)

Answer 5

simplify

Answer 6

but it is 99

Answer 7

@ganeshie8

Answer 8

if 99 divides a number, then 9 also has to divide that same number right ?

Answer 9

so, we're not doing anything illegal.. go ahead setup the equation !

Answer 10

19 + P + Q =9k

Answer 11

yes, we need another equation

Answer 12

since 99 = 9*11,
lets use divisibility rule for 11

Answer 13

setup another equation

Answer 14

so 11k right?

Answer 15

19 + P + Q =11k ?

Answer 16

\(2-1+3-P+4-3+Q-6 = 11k \)

Answer 17

http://www.mathsisfun.com/divisibility-rules.html

Answer 18

you need to take alternating sum for 11

Answer 19

oh so i will simplify wait

Answer 20

-1 - P + Q = 11k

Answer 21

yes, so we have two equations :


1 + P + Q =9k
-1-P+Q = 11m

Answer 22

yes but how to solve them simultaneously we have three variables

Answer 23

easy, we dont solve the varibales

Answer 24

just see that P and Q are just "digits"
that means they have just SINGLE CHARACTER right ?

they have to be between 0-9

Answer 25

yes

Answer 26

1 + P + Q =9k
-1-P+Q = 11m

Answer 27

look at first equation,
we want "1 + P + Q" to be multiple of 9

Answer 28

that means, P+Q can be 8 or 17

Answer 29

youw wid me ?

Answer 30

how 17

Answer 31

1 + P + Q =9k

plugin k = 1, 2, ...

Answer 32

we dont want values from k=3 because they make each of P or Q greater than 9.

Answer 33

ok i got it

Answer 34

P + Q = 17
P + Q = 8

Answer 35

^ these are the equations

Answer 36

# 1 + P + Q =9k
you got :
P + Q = 17
P + Q = 8


#-1-P+Q = 11m
you got :
??

Answer 37

find the equations for 11m also

Answer 38

How many numbers to put in because it is -P

Answer 39

good question,
max value of any digit = 9
min value of any digit = 0
so the difference should not exceed 9

Answer 40

-1-P+Q = 11m

put m = 0

Answer 41

Q-P=1

Answer 42

So, it all boils down to solving :

# 1 + P + Q =9k
you got :
P + Q = 17
P + Q = 8


#-1-P+Q = 11m
you got :
-P + Q = 1

Answer 43

take the first equation from first group
take the first equation from second group

solve them

Answer 44

P+Q = 17
-P + Q = 1

Answer 45

wat do u get when u solve them ?

Answer 46

(8,9)

Answer 47

bingo ! thats ur one solution !
lets see if we can get one more solution or not

Answer 48

we won't

Answer 49

take the second equation from first group
take the first equation from second group

and solve them

Answer 50

how do u say that ?

Answer 51

P + Q = 8
-P + Q = 1

Answer 52

solve them

Answer 53

Because the answer is only (P,Q)

Answer 54

I mean it is given in the book

Answer 55

solve them and see wat u get ?

Answer 56

But logically it seems correct to solve the equations but we don't get whole numbers

Answer 57

exactly ! you got it !!

Answer 58

we don't get whole numbers

Answer 59

u genius man!! india se hai kya

Answer 60

for below equations :
P + Q = 8
-P + Q = 1

we wont get whole numbers for P and Q, so this is not a valid solution. throw it !

Answer 61

only one ordered pair exists (P, Q) = (8, 9)

Answer 62

Good problem thank u you

Answer 63

lol yes india... but no genius lol

Answer 64

you're the genius :)

Answer 65

kidhar mumbai me rehta hai

Answer 66

bangalore, wbu ?

Answer 67

mumbai

Answer 68

what is ur name?

Answer 69

ganesh tadi i got it

Answer 70

ganesh yes :) btw wat u studying ?

Answer 71

ahh ur profile is empty, wats ur name ?

Answer 72

I am preparing for IIT . I passed tenth last year in november it is because i am from different board not sse. So i have joined resonance this year and my entrance test is on 30 th march .

Answer 73

Udit kulkarni

Answer 74

So u an IIT

Answer 75

i mean iitian

Answer 76

omg! i thought you were atleast 12th + ! lol you're too smart !!

Answer 77

nope, and im done wid my studies, been working from > 5 years actually...

Answer 78

So what is IC engineeing like never heard of it actually before

Answer 79

gtg... need to get ready for work...
good luck wid entrance exam :) im sure u wil do great !!
cya :)

Answer 80

chip designing

Answer 81

you must have heard about "integrated circuits" ?

Answer 82

Thank you and wish u luck too!

Answer 83

yes

Answer 84

cool!!

Answer 85

IC design involves designing those chips
wil talk more ltr byeee have a great day !

Answer 86

same 2 u bye