Question

Let S be the sum of all two digit numbers each of which contains one odd and one even digits. Then S is divisible by

(a) 2
(b) 3
(c) 5
(d) 7

There may be more than 1 correct answer

Answer 1

myninaya can u help me ?

Answer 2

S=x+y
x=10n+c
y=10d+k
since x and y are two digit numbers
n,d,c, and k are the digits of x and y
n is the 10's digit for x
c is the 1's digit for x
d is the 10's digit for y
k is the 1's digit for y

ok still thinking...

Answer 3

ok
let me read this

Answer 4

ok so n or c will be even but not both
and
d or k will be even but not both

Answer 5

lets just assume for right n is even
so n=2t
so we have x=10(2t)+c=20t+c
so c is odd so c=2s+1
x=20t+(2s+1)=20t+2s+1=2(10t+s)+1
so x is odd

Answer 6

ok

Answer 7

ok lets assume d is even
so d=2j
then k=2g+1
so
y=10(2k)+2g+1=2(10k+g)+1
so y is odd
so S=x+y=2(10t+s)+1+2(10k+g)+1=2(10t+s)+2+2(10k+g)=2(10t+s+1+10k+g)
so S is even and therefore 2 goes into S

Answer 8

I have the answer of this question but can u explain it

Answer 9

ok

Answer 10

it is certainly divisible by 2

Answer 11

yes which i showed above

Answer 12

with alot of different variables lol

Answer 13

heh, you needn't have done all that to prove it, but it is formal math.

Answer 14

wait, is 0 considered even?

Answer 15

0 is even

Answer 16

i define anything divisble by 2 to be even

Answer 17

in the number 10, 1 is odd, but is 0 even?
and are we allowed to consider the numbers 10,20,30 and so on?

Answer 18

sorry, not 20

Answer 19

I have the answer of this question but can u explain it

Answer 20

The sum of all two digit numbers = 10 + 11 + 12 + .... + 99

= 99(10 + 99)/2 = 4905

There are 5 x 5 = 25 numbers with two digits. The sum of these numbers = 5(1+3+5+7+9) x 11 = 55 x 25 = 1375.

Likewise, th sum of numbers with two even digits
= 5(0+2+4+6+8) x 11 - (0+2+4+6+8)

= 54 x 20

= 1080

Therefore, S = 4905 - 1375 = 2450 = \[2.5^2.7^2\]

Therefore, correct answers are (a),(c) and (d).

Answer 21

I have the answer of this question but can u explain it

Answer 22

Now will you please explain it.

Answer 23

wow never seen a question like i dont believe or a solution like that i mean but let me look at it

Answer 24

dhat do you understand it

Answer 25

I have the answer of this question but can u explain it

Answer 26

what do you mean to say myininaya ?
I can't understand what u mean to say.

Answer 27

no, not really. I am trying to read it carefully

Answer 28

I have the answer of this question but can u explain it

Answer 29

ok

Answer 30

I have the answer of this question but can u explain it

Answer 31

Take ur time

Answer 32

I understnd that there are 25 numbers with both digits odd and 20 numbers with both even digits. but I don't understand how they came up with their sum

Answer 33

I have the answer of this question but can u explain it

Answer 34

do u mean this step

= 90/2 ( 10+ 99) ?

I know how it came

Answer 35

I have the answer of this question but can u explain it

Answer 36

by applying the formula of sum of arithmetic progression

Answer 37

no not that. i meant how they came up with 1375 and 1080

Answer 38

I have the answer of this question but can u explain it

Answer 39

sorry that i also don't know

Answer 40

11+13+15+17+19+31+33+35+37+39+51+53+55+57+59+71+73+75+77+79+91+93+95+97+99

Answer 41

if you add all that up you get 1375. but, I don't understand the formula you gave for that. there must be a pattern we are missing

Answer 42

I have the answer of this question but can u explain it

Answer 43

u mean

5(1 + 3 + 5 + 7 +9 ) * 11

Answer 44

yes

Answer 45

I have the answer of this question but can u explain it

Answer 46

i don't know sry

Answer 47

I have the answer of this question but can u explain it

Answer 48

bhaiya batao naa please'

Answer 49

well, empirically, the answer is correct. you can add up all the numbers I provided and end up with 1375
similarly if you add up
20+22+24+26+28+40+42+44+46+48+60+62+64+66+68+80+82+84+86+88
you get 1080.
use your calculator.
but this is not an elegant method. you are using brute force.

Answer 50

maybe amistre can help
i will give you a medal amistre :)

Answer 51

Let S be the sum of all two digit numbers each of which contains one odd and one even digits. is that "at least one odd or one even"? or only one odd and one even?

Answer 52

I have the answer of this question but can u explain it

Answer 53

amisre will u ?

Answer 54

I have the answer of this question but can u explain it

Answer 55

no only 1 odd and 1 even

Answer 56

i dont like that solution and i cant explain it :(

Answer 57

anyway, if the sum of all numbers between 10 and 99 is 4905
and the sum of all numbers with both digits odd is 1375
and the sum of all numbers with both digits even is 1080,
then the sum of all numbers with one odd digit and one even digit has to be 4905-1080-1375 = 2450
which is divisible by 2,5 and 7.

Answer 58

1,3,5,7,9; 2,4,6,8 is 0 even or odd or neither?

Answer 59

I have the answer of this question but can u explain it

Answer 60

in my view it is even

Answer 61

0 is even

Answer 62

5*5 = 25 number that are odd then even

4*5 = 20 numbers that are even then odd right?

Answer 63

I have the answer of this question but can u explain it

Answer 64

right but how do u get that ?

Answer 65

oh i misread the question lol
i guess that could be my problem

Answer 66

that many you start with how many options you get to start with; then each option has the same amount of branches from it that equal the second set of options

5 odd branches with 5 even fingers each = 25 choices

Answer 67

I have the answer of this question but can u explain it

Answer 68

k

Answer 69

4 even branches with 5 odd fingers = 20 choices

45 numbers total in the pool

Answer 70

I have the answer of this question but can u explain it

Answer 71

yaaaaa

Answer 72

12
14
---
26
16
---
42
18
---
60

12, 26, 42, 60, 80,

B{n} = B{n-1} +B{n-2}

B{1} = 12
B{2} = 12 + 12 +2
B{3} = 12 + 14 + 12 +4
B{4} = 12 + 14 + 16 +12+6

B{n} = B{n-1} + B{0}+2(n-1) might be helpful

Answer 73

B{1} not B{0} :)

Answer 74

B{n} = B{n-1} +B{1} +2(n-1)

B{n-1} = B{n-2} +B{1} +2(n-2)

B{n-2} = B{n-3} +B{1} +2(n-3)

B{n} = [[B{n-3} +B{1} +2(n-3)] +B{1} +2(n-2)] +B{1} +2(n-1)
= B{n-3} +3B{1} +2(3n -3 -2 -1)

= B{n-r} +rB{1} +2(rn -(r)-(r-1)-(r-2)-...-3-2-1)

n-r = 1 when r = n-1


B{n} = B{n-(n-1)} +(n-1)B{1} +2((n-1)n -(n-1)-(n-1-1)-(n-1-2)-...-3-2-1)
= 12 +(n-1)(12) +2(n^2-n -n+1-n+2-n+3-...-3-2-1)
= 12(n) +2(n^2-n + A)

A=(-n+1)+(-n+2)+(-n+3)+...+ (-3) + (-2) + (-1) )
A= (-1) + (-2) + (-3) +...+(-n+3)+(-n+2)+(-n+1))
-------------------------------------------------
2A= -n + -n + -n +...+ -n + -n + -n

A = -n(....)/2

Answer 75

1 | 0 2 4 6 8
2 | 1 3 5 7 9
3 | 0 2 4 6 8
4 | 1 3 5 7 9
5 | 0 2 4 6 8
6 | 1 3 5 7 9
7 | 0 2 4 6 8
8 | 1 3 5 7 9
9 | 0 2 4 6 8

Let M be the sum of all two digit numbers
M=10+11+12+....+99=(1+2+3+....+9)+10+11+12+13+...+99-(1+2+3+...+9)
=99(99+1)/2-45=4950-45=4905

Look at the chart above .
the only thing that is missing are: 11,13,15,17,19,20,22,24,26,28,31,33,35,37,39,40,42,44,46,48,51,53,55,57,59,60,62,
64,66,68,71,73,75,77,79,80,82,84,86,88,91,93,95,97,99

now we can separate these numbers into two groups

both even digits:={20,22,24,26,28,40,42,44,46,48,60,62,64,68,80,82,84,86,88}

both odd digits:={11,13,15,17,19,31,33,35,37,39,51,53,55,57,59,71,73,75,77,79,91,93,95,97,99}



the sum of the elements of the both even digit set is =20+22+24+26+28+40+42+44+46+48+60+62+64+68+80+82+84+86+88
=1080
the sum of the elements of the both odd digit set is
=11+13+15+17+19+31+33+35+37+39+51+53+55+57+59+71+73+75+77+79+
91+93+95+97+99=1375


so S can be found by doing S=M-sum of both even digits-sum of both odd digits
=4905-1080-1375=2450

we should write 2450 in prime factorization to better see its factors
2450=2*5^2*7^2

Answer 76

B{n} = 12(n) +2n^2 -n(n+1)

12 = 12(1) + 2 - 1(2)

26 = 12(2) +2(4) - 2(3)
= 24 + 8 - 6

42 = 12(3) +2(9) -3(4)
= 36 + 18 -12
= 36 + 6

60 = 12(4) + 2(16) -4(5)
= 48 + 32 - 20
= 48 + 12

think I got it :) at least part of it lol

Answer 77

:)

Answer 78

70 + 170 + 270 + 370 + 470 = 1350 for the odds ... maybe :)

Answer 79

had to start at 10; 30; 50; 70; and 90

Answer 80

2 2 2
21 41 61
23 43 63
25 45 65
27 47 67
29 49 69
----- ---- ----
125 225 325 425 = 1000 + 1350 = 2350 maybe?

Answer 81

omg almost

Answer 82

2 2
10 30
12 32
14 34
16 36
18 38
--- ----
70 170 270 370 470 = 1350 right :)

Answer 83

1100 + 1350 = 2450 :)

Answer 84

yes!:)

Answer 85

great amistre and myininaya u made a good effort to solve my brother's question

Answer 86

thanks from my elder brother's side he would reply later

Answer 87

and also dhatraditya thanks