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Remainder

05/23/2008 5:55 AM

I just found a number with an interesting property:

When I divide it by 2, the remainder is 1.
When I divide it by 3, the remainder is 2.
When I divide it by 4, the remainder is 3.
When I divide it by 5, the remainder is 4.
When I divide it by 6, the remainder is 5.
When I divide it by 7, the remainder is 6.
When I divide it by 8, the remainder is 7.
When I divide it by 9, the remainder is 8.
When I divide it by 10, the remainder is 9.

It's not a small number, but it's not really big, either.

Find the smallest number with such property.

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#1

Re: Remainder

05/23/2008 8:02 AM

Aaaaaaargh! This one is worse that PlbMak's Friday afternoon picture quizzes!

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#2

Re: Remainder

05/23/2008 8:03 AM

Coz I a Cat, I dun do good with numbery things (they are tooo wriggly) but I wurked out the least significant digit is 9... can I go home now

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#3

Re: Remainder

05/23/2008 8:22 AM

5039

Now I can go to sleep .

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#4

Re: Remainder

05/23/2008 9:14 AM

Hi Optimusprime,

It's 2519

Mike

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#5

Re: Remainder

05/24/2008 12:48 AM
abcd
279251925192519
279251925191259 1/2
27925192519839 2/3
27925192519629 3/4
27925192519503 4/5
27925192519419 5/6
27925192519359 6/7
27925192519314 7/8
27925192519279 8/9

'excel' helped me; b=9a+8 and went on changing 'a' with numbers ending with 9.

any short approach?

subhu, India

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#6

Re: Remainder

05/24/2008 1:34 AM

yes... 2519,5039,7559,10079...and the list goes on

Try same with 2521,5041,7561 &10081

Well I used the function MOD in Excel..Of course I could have written a small program with loops in loops... I believe that is the way to do it. Once you write the program you can change the conditions they way you want it and create many puzzles.

BTW Optimusprime I noted in one of your other posts about UV protection for old . It would be interesting to find out what techniques they use to solve this.

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#7
In reply to #6

Re: Remainder

05/24/2008 7:54 AM

If you're waiting for Optimusprime to answer, he/she probably won't. He/she has asked a lot of questions, but never has posted! Talk about a one-way conversation.

I did this in Excel and got the same answer (2519) by brute force using the MOD function. I thought about writing a small program too, but just kept entering prime numbers with LSD (that's Least Significant Digit) = 9. It would have been faster to write a program!

Mike

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#8
In reply to #6

Re: Remainder

05/24/2008 8:38 AM

If you're waiting for Optimusprime to answer, he/she probably won't. He/she has asked a lot of questions, but never has posted! Talk about a one-way conversation.

Oops, never mind that first paragraph - I misspoke.

Mike

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#9

Re: Remainder

05/25/2008 2:56 AM

The smallest number is 1.

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#10

Re: Remainder

05/25/2008 4:09 PM

Optimusprime,

Good question! I agree that the answer is 2519. It is one less than the product of all the prime factors for all the divisors.

Divisor = 2; prime factor is 2; number is 2-1 = 1.

Divisors = 2,3; prime factors are 2,3; number is 2*3-1 = 5.

Divisors = 2,3,4; prime factors are 2,2,3; number is 2*2*3-1 = 11.

Divisors = 2,3,4,5; prime factors are 2,2,3,5; number is 2*2*3*5-1 = 59.

Divisors = 2,3,4,5,6; prime factors are 2,2,3,5; number is 2*2*3*5-1 = 59. (same #!)

Divisors = 2,3,4,5,6,7; prime factors are 2,2,3,5,7; number is 2*2*3*5*7-1 = 419.

Divisors = 2,3,4,5,6,7,8; prime factors are 2,2,2,3,5,7; number is 2*2*2*3*5*7-1 = 839.

Divisors = 2,3,4,5,6,7,8,9; prime factors are 2,2,2,3,3,5,7; number is 2*2*2*3*3*5*7-1 = 2159.

Divisors = 2,3,4,5,6,7,8,9,10; prime factors are 2,2,2,3,3,5,7; number is also 2,159.

Now, you can see the general relationship. To extend this further, if the next two divisors are 11 and 12, the number will be 2*2*3*3*5*7*11-1 = 27,719. For divisors of 13, 14, and 15, the number will be 360,359.

Now a return question: What will the number be when you first have four divisors in the sequence which all work on the same number? (Above, you saw three instances where two divisors in sequence both worked on the same number and then one where three in sequence worked on the same number.)

--John M.

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#12
In reply to #10

Re: Remainder

05/27/2008 9:51 AM

..19,..20,..21 and ..22 unless I've misunderstood the question.

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#11

Re: Remainder

05/27/2008 7:34 AM

First notice that -1 (minus one) satisfies the requirements (if not the implied one). Then notice that adding the Least Common Multiple as many times as you want gives all the answers. So add the LCM once to give the smallest number. As others have already pointed out the LCM is 2520 so the answer is 2519.

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#13

Re: Remainder

05/28/2008 10:27 AM

I make it 2519. But I wouldn't have managed it without XL! (that's assuming it's right, and I haven't looked at other posts yet)

Codey

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