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# The numbers 400, 536 and 645, when divided by a positive integer N, gi

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Math Expert
Joined: 02 Sep 2009
Posts: 64133
The numbers 400, 536 and 645, when divided by a positive integer N, gi  [#permalink]

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24 Feb 2020, 04:47
00:00

Difficulty:

55% (hard)

Question Stats:

67% (02:23) correct 33% (02:53) wrong based on 52 sessions

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The numbers 400, 536 and 645, when divided by a positive integer N, give the remainders of 22, 23 and 24 respectively. What is the greatest possible value of N?

A. 9
B. 18
C. 21
D. 27
E. 54

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Re: The numbers 400, 536 and 645, when divided by a positive integer N, gi  [#permalink]

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24 Feb 2020, 05:04
1
METHOD-1
N will be the HCF of (400 – 22), (536 – 23) and (645 – 24). Hence, N will be the HCF of 378, 513 and 621. N = 27.
Time taken to solve : 60secs

METHOD-2
Individually use the options to find the answer. It will take around 120secs to 180secs
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Re: The numbers 400, 536 and 645, when divided by a positive integer N, gi  [#permalink]

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29 Mar 2020, 06:29
Bunuel wrote:
The numbers 400, 536 and 645, when divided by a positive integer N, give the remainders of 22, 23 and 24 respectively. What is the greatest possible value of N?

A. 9
B. 18
C. 21
D. 27
E. 54

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400modN = 22
536modN = 23
645modN = 24

136modN=1
109modN=1

27modN=0
N = 27k

Let N = 54;
400mod54= 22
536mod54 = -4mod54 = 50
645mod54 = -3mod54 = 51

Largest value of N = 27

IMO D
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Re: The numbers 400, 536 and 645, when divided by a positive integer N, gi  [#permalink]

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29 Mar 2020, 07:34
1
Bunuel wrote:
The numbers 400, 536 and 645, when divided by a positive integer N, give the remainders of 22, 23 and 24 respectively. What is the greatest possible value of N?

A. 9
B. 18
C. 21
D. 27
E. 54

400 divided by N gives remainder 22
i.e. 400-22 = 378 is divisible by N

536 divided by N gives remainder 23
i.e. 536-23 = 513 is divisible by N

645 divided by N gives remainder 24
i.e. 645-24 = 621 is divisible by N

i.e. N must be Common Factor of {378, 513 and 621}.

Option B and E are out as factor of an Odd number must be ODD

Sum of digits of all Numbers is divisibleby 9 i.e. N must be divisible by 9

Option C is out as they are not multiples of 9

But all of them are divisible by 27 hence

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Re: The numbers 400, 536 and 645, when divided by a positive integer N, gi  [#permalink]

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29 Mar 2020, 07:46
The numbers 400, 536 and 645, when divided by a positive integer N, give the remainders of 22, 23 and 24 respectively. What is the greatest possible value of N?

A. 9
B. 18
C. 21
D. 27 --> correct
E. 54

Solution:
400 = N*d1 + 22 => N*d1=378 = 3^3*14 ---(i)
536 = N*d2 + 23 => N*d2=513 = 3^3*19---(ii)
645 = N*d3 + 24 => N*d3=621 = 3^3 *23--(iii)

HCF of (i), (ii) & (iii) => 3^3=27

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Re: The numbers 400, 536 and 645, when divided by a positive integer N, gi  [#permalink]

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29 Mar 2020, 12:23
1

Solution

Given
In this question, we are given that
• The numbers 400, 536 and 645, when divided by a positive integer N, give the remainders of 22, 23 and 24 respectively

To find
We need to determine
• The greatest possible value of N

Approach and Working out
When 400 is divided by N, the remainder is 22
• Hence, 400 – 22 = 378 should be completely divisible by N

When 536 is divided by N, the remainder is 23
• Hence, 536 – 23 = 513 should be completely divisible by N

When 645 is divided by N, the remainder is 24
• Hence, 645 – 24 = 621 should be completely divisible by N

Now, the greatest possible value of N = the greatest number that divides 378, 513, 621 = GCD (378, 513, 621) = 27

Thus, option D is the correct answer.

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Re: The numbers 400, 536 and 645, when divided by a positive integer N, gi  [#permalink]

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29 Mar 2020, 14:02
easiest way to solve this problem is by substracting remainder from given number for example do 400-22 =388 and then match from the options.
Re: The numbers 400, 536 and 645, when divided by a positive integer N, gi   [#permalink] 29 Mar 2020, 14:02