Puzzle for February 27, 2021 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit positive integer.
* BC is a 2-digit number (not B×C).
** "F mod B" is the remainder of (F ÷ B). "F mod E" is the remainder of (F ÷ E).
Scratchpad
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Hint #1
Subtract the left and right sides of eq.3 from the left and right sides of eq.2, respectively: B + D – (D + F – B) = C + F – (B + C) which is equivalent to B + D – D – F + B = C + F – B – C which becomes 2×B – F = F – B Add F and B to both sides of the above equation: 2×B – F + F + B = F – B + F + B which makes 3×B = 2×F Divide both sides by 2: 3×B ÷ 2 = 2×F ÷ 2 which makes eq.2a) 1½×B = F
Hint #2
In eq.5, replace F with 1½×B: 1½×B mod B = D – B which means remainder of (1½×B ÷ B) = D – B which means ½×B = D – B Add B to both sides of the above equation: ½×B + B = D – B + B which makes 1½×B = D and therefore makes F = D = 1½×B
Hint #3
In eq.2, substitute 1½×B for D and F: B + 1½×B = C + 1½×B Subtract 1½×B from each side of the above equation: B + 1½×B – 1½×B = C + 1½×B – 1½×B which makes B = C
Hint #4
eq.4 may be written as: 10×B + C = (D × E) – C Substitute B for C, and 1½×B for D in the above equation: 10×B + B = (1½×B × E) – B which becomes 11×B = (1½×B × E) – B Add B to both sides: 11×B + B = (1½×B × E) – B + B which becomes 12×B = 1½×B × E Divide both sides by 1½×B: 12×B ÷ 1½×B = 1½×B × E ÷ 1½×B which makes 8 = E
Hint #5
Substitute 8 for E, and B for C in eq.6: F mod 8 = B ÷ B which means remainder of (F ÷ 8) = 1 Since F is a one-digit integer, the above equation makes either: F = 1 or F = 9
Hint #6
Check: F = 1 ... Substituting 1 for F into eq.2a would yield: 1½×B = 1 Dividing both sides of the above equation by 1½ would yield: 1½×B ÷ 1½ = 1 ÷ 1½ which would make B = ⅔ Since B is an integer, then B ≠ ⅔ which means F ≠ 1 and, therefore makes F = 9 which also makes D = F = 9
Hint #7
Substitute 9 for F in eq.2a: 1½×B = 9 Divide both sides of the above equation by 1½: 1½×B ÷ 1½ = 9 ÷ 1½ which makes B = 6 and also makes C = B = 6
Solution
Substitute 6 for B and C, 9 for D and F, and 8 for E in eq.1: A + 6 + 6 + 9 + 8 + 9 = 39 which becomes A + 38 = 39 Subtract 38 from each side of the equation above: A + 38 – 38 = 39 – 38 which makes A = 1 and ABCDEF = 166989