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20 Dec 2024, 09:00
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Fourth digit: 5
Fifth digit: 4
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
74% (03:07) correct
26%
(03:19)
wrong
based on 677
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12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes
Myrna, Deon, and Kermit just moved into a new house together and each has made an observation about the numbers in their new five-digit address. Myrna has observed that the first two digits sum to the third digit and that the last two digits also sum to the third digit. Deon has observed that the third digit is a factor of both the two-digit number formed by the first two digits and the two-digit number formed by the last two digits. Kermit has observed that the values of the digits increase from the first to the third digit and then decrease from the third to the fifth digit.
If all of the digits in the address are unique, select from the available options a fourth digit and a fifth digit for the address that are jointly consistent with the information given. Make only two selections, one in each column.
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Myrna, Deon, and Kermit just moved into a new house together and each has made an observation about the numbers in their new five-digit address. Myrna has observed that the first two digits sum to the third digit and that the last two digits also sum to the third digit. Deon has observed that the third digit is a factor of both the two-digit number formed by the first two digits and the two-digit number formed by the last two digits. Kermit has observed that the values of the digits increase from the first to the third digit and then decrease from the third to the fifth digit.
If all of the digits in the address are unique, select from the available options a fourth digit and a fifth digit for the address that are jointly consistent with the information given. Make only two selections, one in each column.
| Fourth digit | Fifth digit | |
| 0 | ||
| 3 | ||
| 4 | ||
| 5 | ||
| 7 | ||
| 8 |
ShowHide Answer
Official Answer
Fourth digit: 5
Fifth digit: 4
20 Dec 2024, 10:00
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Bunuel
Manhattan Prep Official Explanation:
Step 1: Understand the Prompt and Question
Glance at the answers and then the prompt. The numbers in the answers and the references to digits and factors in the prompt suggest this is a Quant-based question testing Number Properties, specifically Divisibility & Primes. Next, read the entire prompt carefully and jot down the given information on your paper:
5-digit address: _a_ _b_ _c_ _d_ _e_
a + b = d + e = c
c is a factor of the 2-digit #’s a b and d e
a < b < c and c > d > e
all 5 digits ≠
Step 2: Plan your Approach
The limited number of possibilities for unique digits and the many constraints given suggest that Testing Cases could be a possibility. Since you are asked to find fourth and fifth digits that are jointly consistent with given information, once you find a pair of answers that work, those must form the correct answer.
Step 3: Solve the Problem
Testing the middle digit c as 9 seems most promising, as many digit pairs will sum to 9: {1, 8}, {2, 7}, {3, 6}, and {4, 5}. Moreover, any 2-digit number with digits that sum to 9 will be divisible by 9, ensuring the constraint that c is a factor of both a b and d e is met.
Among the answer choices, only the pair {4, 5} can be found, and given the constraint c > d > e the fourth and fifth digits would be 5 and 4, respectively. The full address could be any of the following 18954, 27954, or 36954.
The correct answer is 5 for the first column (the Fourth digit) and 4 for the second column (the Fifth digit).
General Discussion
20 Dec 2024, 09:49
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This is how I approached it:
First, we eliminate the digit 0 from the options because adding 0 to another digit simply gives back that digit. This would make the third digit the same as either the fourth or fifth digit, which is not allowed because all digits in the address must be unique.
The digits 7 and 8 are also eliminated because no other digit (apart from 0, which is invalid) can combine with them to form a single-digit sum.
This leaves us with the digits 3, 4, and 5. Possible combinations for the fourth and fifth digits are 43, 53, and 54 (decreasing order constraint) .
Next, we eliminate 53 because it is a prime number and cannot have 9 (the sum of 5 and 4) as a factor.
Similarly, 43 is eliminated because it is also a prime number and does not have 7 (the sum of 4 and 3) as a factor.
This leaves us with 54 as the only valid option. The sum of 5 and 4 is 9, and 9 is a factor of 54, satisfying all the constraints.
With the fourth and fifth digits as 5 and 4, the full address must also ensure that the first two digits sum to 9 (the third digit) and that all digits follow the increasing-decreasing pattern.
Valid complete addresses are 18954, 27954, and 36954, as they satisfy all conditions:
So, the fourth digit is 5, and the fifth digit is 4.
First, we eliminate the digit 0 from the options because adding 0 to another digit simply gives back that digit. This would make the third digit the same as either the fourth or fifth digit, which is not allowed because all digits in the address must be unique.
The digits 7 and 8 are also eliminated because no other digit (apart from 0, which is invalid) can combine with them to form a single-digit sum.
This leaves us with the digits 3, 4, and 5. Possible combinations for the fourth and fifth digits are 43, 53, and 54 (decreasing order constraint) .
Next, we eliminate 53 because it is a prime number and cannot have 9 (the sum of 5 and 4) as a factor.
Similarly, 43 is eliminated because it is also a prime number and does not have 7 (the sum of 4 and 3) as a factor.
This leaves us with 54 as the only valid option. The sum of 5 and 4 is 9, and 9 is a factor of 54, satisfying all the constraints.
With the fourth and fifth digits as 5 and 4, the full address must also ensure that the first two digits sum to 9 (the third digit) and that all digits follow the increasing-decreasing pattern.
Valid complete addresses are 18954, 27954, and 36954, as they satisfy all conditions:
- The sum of the first two digits equals the third digit
- The sum of the last two digits equals the third digit
- The third digit is a factor of the two-digit numbers formed by the first two and the last two digits
- The digits increase to the third digit and decrease afterward
So, the fourth digit is 5, and the fifth digit is 4.
