#1

How many positive integers less than 10001000 have the property that the sum of the digits of each such number is divisible by 77 and the number itself is divisible by 33?

#NumberTheory

Note by Vilakshan Gupta
3 years, 9 months ago

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Comments

Let's think of a number abc(0a,b,c9)abc (0 \leq a, b, c \leq 9). a+b+c0(mod3andmod7)a+b+c \equiv 0 (\mod 3 \text{and} \mod 7). Thus, a+b+c=21a+b+c=21.

(3,9,9)3!2,(4,8,9)3!,(5,7,9)3!,(5,8,8)3!2,(6,6,9)3!2,(6,7,8)3!,(7,7,7)(3, 9, 9) \rightarrow \frac{3!}{2}, (4, 8, 9) \rightarrow 3!, (5, 7, 9) \rightarrow 3!, (5, 8, 8) \rightarrow \frac{3!}{2}, (6, 6, 9) \rightarrow \frac{3!}{2}, (6, 7, 8) \rightarrow 3!, (7, 7, 7) 3+6+6+3+3+6+1=283+6+6+3+3+6+1=28

Please tell me if there is any error.

Pepper Mint - 3 years, 7 months ago

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Nice method

Vilakshan Gupta - 3 years, 7 months ago

Good one brother

Utkarsh Kumar - 3 years, 7 months ago

Your method is quite efficient vis-a-vis mine. The latter involved manual trials with 3 digit integers with integer 1 to 9 at the unit. Thank you

Mrigaank Gupta - 1 year, 10 months ago

What ans did you get?

Aaron Jerry Ninan - 3 years, 9 months ago

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Hey Aaron. How much are u getting with bonus? With bonus i am getting 12.

Md Zuhair - 3 years, 9 months ago

25 is the answer as per me

Pokhraj Harshal - 3 years, 9 months ago

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No - I get 28 too - I constructed a 0 - 9 by 0 -9 addition table in excel, and then started adding a 3rd digit to any number whose 2 digits had added to 12 or more - though now I think about it, I could just as easiky have srated my list with 399 and continued from there. And it has to be 28 cos it's one starting with3, 2 starting with 4, 3 starting with 5 etc, and 1+2+3+4+5+6+7 = 28

Katherine barker - 3 years, 9 months ago

28

Vilakshan Gupta - 3 years, 9 months ago

we just need to find the numbers which add upto 21

Vilakshan Gupta - 3 years, 9 months ago

Answer is 28

Toshit Jain - 3 years, 9 months ago

How mañy have you got right in PRMO - 17?

Toshit Jain - 3 years, 9 months ago

@Md Zuhair Is the paper for 9,10,11 and 12 same?

Toshit Jain - 3 years, 9 months ago

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Yes Sir!

Md Zuhair - 3 years, 9 months ago

unfortunately, i will get only 10 questions correct. I did very silly mistakes

Vilakshan Gupta - 3 years, 9 months ago

Sir😅 I am getting 10 along with bonus!

Toshit Jain - 3 years, 9 months ago

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Oh. U mean 8/28 u r getting?

Md Zuhair - 3 years, 9 months ago

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well, if it is bonus that means 2 questions marks are given extra.If it was been written question deleted then scores would be evaluated out of 28

Vilakshan Gupta - 3 years, 9 months ago

Geometry was quite tough and lengthy! Excluding bonus , I'm getting 8

Toshit Jain - 3 years, 9 months ago

oh...btw,where do u live (i mean which region)

Vilakshan Gupta - 3 years, 9 months ago

Rajasthan..U?

Toshit Jain - 3 years, 9 months ago

oh

Vilakshan Gupta - 3 years, 9 months ago

So , you are already selected..Great 👍

Toshit Jain - 3 years, 9 months ago

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How already selected?

Md Zuhair - 3 years, 9 months ago

Coz as per cutoff(s) uploaded by Resonance , cutoff in Chandigarh is lower than others ( Rajasthan , Maharashtra , UP , etc) ... That's why!

Toshit Jain - 3 years, 9 months ago

According to Resonance , cutoff in Chandigarh is just 4(questions)

Toshit Jain - 3 years, 9 months ago

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Ya. Thats ridiculous. WB region has always got a higher cutoff...

Md Zuhair - 3 years, 9 months ago

i don't think it will be so low

Vilakshan Gupta - 3 years, 9 months ago

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If that isnt, then wb will be higher and i will surely not qualify

Md Zuhair - 3 years, 9 months ago

It's 11 in Rajasthan ! 😅😒

Toshit Jain - 3 years, 9 months ago

LetTheFateDecide !!Bye

Toshit Jain - 3 years, 9 months ago

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which class are u in toshit?

Vilakshan Gupta - 3 years, 9 months ago

@Shreyan Chakraborty .. How much?

Md Zuhair - 3 years, 9 months ago

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JANI NA BAJE HOYECHE

Shreyan Chakraborty - 3 years, 9 months ago

ANSWER IS 28....HAS A BIJECTION WITH a+b+c=21 WHERE 0<a,b,c<=9........

Shreyan Chakraborty - 3 years, 9 months ago

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Are na na.... I am not telling that. How much are you getting?

Md Zuhair - 3 years, 9 months ago

@Shreyan Chakraborty The Hundreds digit can't be 1 or 2..

Toshit Jain - 3 years, 9 months ago

yeah hundreds digit cant be 1,2

Shreyan Chakraborty - 3 years, 9 months ago

@Md Zuhair @Vilakshan Gupta I haven't attempted one of the bonus question. Will I still get marks for it?

Toshit Jain - 3 years, 9 months ago

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I think the question can be cancelled as how can a person attempt to decinal answers and i had attempted ine. So i dunno.

Md Zuhair - 3 years, 9 months ago

Yup.I think so.

Vilakshan Gupta - 3 years, 9 months ago

@Vilakshan Gupta Ok!

Toshit Jain - 3 years, 9 months ago

I believe the answer is 28 integers. The sum of these integers' digits must be divisible by 21, since a number divisible by 3 also has its sum of digits divisible by 3; in addition to the sum of digits divisible by 7. None of the digits can be less 3 since the sum of digits would be less than 21. Possible combinations = 7+6+5+4+3+2+1 = (7+1)+(6+2)+(5+3)+4=3*8+4=28.

Simon Dancose - 3 years, 9 months ago

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Exactly

Vilakshan Gupta - 3 years, 9 months ago

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I agree with all that, and I got the same answer, but if I give you 4 digits at random (say 3, 4, 5 and 6) and ask how many numbers you can make out of them, the answer is 432*1 = 24, not 4+3+2+1 = 10. What am I missing?

Katherine barker - 3 years, 9 months ago

Ah - that's where the 28 comes from - much more mathematical than my just listing and counting them

Katherine barker - 3 years, 9 months ago

Hello, There are 28 postive integers left less than 1000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3 You can check out for more queries related to the JEE EXAMS from the following compilation
<a href="https://scoop.eduncle.com/jee-main-exam-date-notification">JEE MAIN Exam dates and Notification</a>

Rahul Sharma - 3 years, 9 months ago

Hey I'm getting 8/27 from jharkhand as per the new answer key of hbcse.will I qualify???

Pokhraj Harshal - 3 years, 9 months ago

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Lets see....

Md Zuhair - 3 years, 9 months ago

Hey , what does discounted actually refer to?

Toshit Jain - 3 years, 9 months ago

Are the marks gonna be added to everyone's total or the questions will be cancelled (lowering the cutoff)?

Toshit Jain - 3 years, 9 months ago

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They will be added to totsl

Alex Roy - 3 years, 8 months ago

The questions will be cancelled

Pokhraj Harshal - 3 years, 9 months ago

@Pokhraj Harshal Ok! Then I'm also getting the same...

Toshit Jain - 3 years, 9 months ago

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How much? Without the question?

Md Zuhair - 3 years, 9 months ago

Which region are u from??

Pokhraj Harshal - 3 years, 9 months ago

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WB rgion

Md Zuhair - 3 years, 9 months ago

And what about the other participants and their marks from your school. I mean the averages and the highest marks

Pokhraj Harshal - 3 years, 9 months ago

@Pokhraj Harshal Rajasthan region!

Toshit Jain - 3 years, 9 months ago

@Md Zuhair 8

Toshit Jain - 3 years, 9 months ago

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O i see....

Md Zuhair - 3 years, 9 months ago

this question came in this year PRMO answer is 28

suresh kumar - 3 years, 8 months ago

its sum is divisibli by 21 using this you can solve

suresh kumar - 3 years, 8 months ago

27

Alex Roy - 3 years, 8 months ago

28

Hemank Bajaj - 3 years, 6 months ago

33

Bimit Mandal - 2 years, 10 months ago

zuhair tui ki amk jiggesh korchish??

Shreyan Chakraborty - 3 years, 9 months ago

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Accha.. nijer whatsapp number ta de... whatsapp e kotha bolchi

Md Zuhair - 3 years, 9 months ago
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