There is a safe with a 5 digit number as the key. The 4th digit is 4 greater than second digit, while 3rd digit is 3 less than 2nd digit. The 1st digit is thrice the last digit. There are 3 pairs whose sum is 11. Find the number.
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Answer / Divya Dixit
Let's denote the digits from left to right as A (first digit), B, C, D, and E (last digit). We know that: 1) D = A/3, 2) C = B - 3, and 3) D = B + 4. Since there are three pairs of digits whose sum is 11, we can list them as follows: a) AB = 5, E = 2 (since the sum of the pair is 7, and D = A/3 implies that A = 3*D = 3*2 = 6, so B = C + 3 = 0 - 3 = -3, and E = D), b) AB = 4, E = 7 (since the sum of the pair is 11, and D = A/3 implies that A = 3*D = 3*7 = 21, so B = C + 3 = 0 - 3 = -3, and E = D), c) AB = 6, E = 5 (since the sum of the pair is 11, and D = A/3 implies that A = 3*D = 3*5 = 15, so B = C + 3 = 0 - 3 = -3, and E = D). We can also note that if AB = 4 or 6, then A cannot be a multiple of 3 since it would imply that E is not equal to D (which contradicts the given information), so A must be 5 in either case. Therefore, the possible key numbers are 50672 and 56572.
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