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A set consists of all three-digit positive integers with the following

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Bunuel
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A set consists of all three-digit positive integers with the following [#permalink] New post   10 Nov 2022, 08:05
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A set consists of all three-digit positive integers with the following properties. Each integer is of the form JKL, where J, K, and L are digits; all the digits are nonzero; and the two-digit integers JK and KL are each divisible by 9. HOW many integers are in the set?

A. 6
B. 7
C. 8
D. 9
E. 10
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gmatophobia
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A set consists of all three-digit positive integers with the following [#permalink] New post   10 Nov 2022, 13:31
The numbers are in the form of JKL and its also given that neither of J, K or L is 0.

Also, JK is divisible by 9, and KL is divisible by 9.

A number is divisible by 9, when the sum of all digits is a multiple of 9.

As the hundreds digit is common let's fix the hundreds digit

J 9 L
J 8 L
J 7 L
J 6 L
J 5 L
J 4 L
J 3 L
J 2 L
J 1 L

In each of the above cases, the value of L and J will be same, and there can only be one value that will satisfy the sum criteria. For ex. in the first case, when 9 is the tens digit, the value of L and J has to be 9. In the second case, when 8 is at the tens place, the value of L and J has to be 1 and so on.

Hence there are 9 cases.

IMO D
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Re: A set consists of all three-digit positive integers with the following [#permalink] New post   11 Nov 2022, 01:38
gmatophobia wrote:
The numbers are in the form of JKL and its also given that neither of J, K or L is 0.

Also, JK is divisible by 9, and KL is divisible by 9.

A number is divisible by 9, when the sum of all digits is a multiple of 9.

As the hundreds digit is common let's fix the hundreds digit

J 9 L
J 8 L
J 7 L
J 6 L
J 5 L
J 4 L
J 3 L
J 2 L
J 1 L

In each of the above cases, the value of L and J will be same, and there can only be one value that will satisfy the sum criteria. For ex. in the first case, when 9 is the hundreds digit, the value of L and J has to be 9. In the second case, when 8 is at the hundreds place, the value of L and J has to be 1 and so on.

Hence there are 9 cases.

IMO D


Since all the digits are non-zero, the answer will be 8 cases right?
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A set consists of all three-digit positive integers with the following [#permalink] New post   11 Nov 2022, 01:40
khushiez wrote:

Since all the digits are non-zero, the answer will be 8 cases right?


999 can be a possible number.
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Re: A set consists of all three-digit positive integers with the following [#permalink] New post   11 Nov 2022, 01:42
gmatophobia wrote:
khushiez wrote:

Since all the digits are non-zero, the answer will be 8 cases right?


999 can be a possible number.


Oh right! Thank you
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Re: A set consists of all three-digit positive integers with the following [#permalink] New post   18 Nov 2022, 23:22
gmatophobia wrote:
The numbers are in the form of JKL and its also given that neither of J, K or L is 0.

Also, JK is divisible by 9, and KL is divisible by 9.

A number is divisible by 9, when the sum of all digits is a multiple of 9.

As the hundreds digit is common let's fix the hundreds digit

J 9 L
J 8 L
J 7 L
J 6 L
J 5 L
J 4 L
J 3 L
J 2 L
J 1 L

In each of the above cases, the value of L and J will be same, and there can only be one value that will satisfy the sum criteria. For ex. in the first case, when 9 is the hundreds digit, the value of L and J has to be 9. In the second case, when 8 is at the hundreds place, the value of L and J has to be 1 and so on.

Hence there are 9 cases.

IMO D


I think you meant " tens " place rather than " hundred's place " in the highlighted portion above?
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A set consists of all three-digit positive integers with the following [#permalink] New post   19 Nov 2022, 01:22
stne wrote:
gmatophobia wrote:
The numbers are in the form of JKL and its also given that neither of J, K or L is 0.

Also, JK is divisible by 9, and KL is divisible by 9.

A number is divisible by 9, when the sum of all digits is a multiple of 9.

As the hundreds digit is common let's fix the hundreds digit

J 9 L
J 8 L
J 7 L
J 6 L
J 5 L
J 4 L
J 3 L
J 2 L
J 1 L

In each of the above cases, the value of L and J will be same, and there can only be one value that will satisfy the sum criteria. For ex. in the first case, when 9 is the hundreds digit, the value of L and J has to be 9. In the second case, when 8 is at the hundreds place, the value of L and J has to be 1 and so on.

Hence there are 9 cases.

IMO D


I think you meant " tens " place rather than " hundred's place " in the highlighted portion above?


Ah ! right. Fixed the typo.
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A set consists of all three-digit positive integers with the following [#permalink]
19 Nov 2022, 01:22
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