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

Question

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

Show Answer

Official Answer

D

A. 6
B. 7
C. 8
D. 9
E. 10

gmatophobia · Answer

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

khushiez · Answer

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

gmatophobia · Answer

999 can be a possible number.

EpicKnot · Answer

JKL is a non-zero interger such that JK and KL are divisible by 9.

Rule: For a number to be divisible by 9, the sum of it's digits must be a multiple of 9.

Also, K is a middle digit and impacts both JK and KL. So we can take it as a linking digit between all three i.e. value of K would impact both J and L.

Step 1: Finding range of JK and KL. All digits are between 1 and 9. So, Smallest number is 2 (1+1) and largest number is 18 (9+9)

Step 2: Let K =1, then J /L can be either 8 (possible) or 17 (not possible since 2 digits). Hence JK = 81 and KL = 18. So, 3 digit number = 818.

Similarly, when k = 2, JK = 72 and KL = 27. So, 3 digit number = 727

When k = 3, 3 digit number = 636 and so on..

When k = 9, JK = 99 and KL = 99, so 3 digit number = 999

Therefore total numbers = 9.

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

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

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