Puzzle for June 1, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
* AB and CD are 2-digit numbers (not A×B or C×D).
Scratchpad
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Hint #1
eq.6 can be re-written as: 10×A + B + F = 10×C + D - F Add F to both sides of the equation above: 10×A + B + F + F = 10×C + D - F + F which becomes 10×A + B + 2×F = 10×C + D which may also be written as eq.6a) 10×A + B + 2×F = 9×C + C + D
Hint #2
Add A to both sides of eq.4: C + D - A + A = A + B + A which becomes eq.4a) C + D = 2×A + B In eq.6a, replace C + D with 2×A + B: 10×A + B + 2×F = 9×C + 2×A + B Subtract both 2×A and B from each side of the equation above: 10×A + B + 2×F - 2×A - B = 9×C + 2×A + B - 2×A - B which becomes eq.6b) 8×A + 2×F = 9×C
Hint #3
Add (C - E) to both sides of eq.5: E + F - C + (C - E) = A + C + (C - E) which becomes eq.5a) F = A + 2×C - E In eq.6b, substitute (A + 2×C - E) for F: 8×A + 2×(A + 2×C - E) = 9×C which is equivalent to 8×A + 2×A + 4×C - 2×E = 9×C Subtract 4×C from both sides: 8×A + 2×A + 4×C - 2×E - 4×C = 9×C - 4×C which becomes eq.6c) 10×A - 2×E = 5×C
Hint #4
In eq.4a, replace B with D + E (from eq.2): C + D = 2×A + D + E Subtract D from both sides of the equation above: C + D - D = 2×A + D + E - D which becomes eq.4b) C = 2×A + E
Hint #5
Substitute (2×A + E) for C (from eq.4b) in eq.6c: 10×A - 2×E = 5×(2×A + E) which becomes 10×A - 2×E = 10×A + 5×E Add 2×E to each side, and subtract 10×A from each side of the equation above: 10×A - 2×E + 2×E - 10×A = 10×A + 5×E + 2×E - 10×A which simplifies to 0 = 7×E which means 0 = E
Hint #6
Substitute 0 for E in eq.2: B = D + 0 which makes B = D
Hint #7
Substitute 0 for E in eq.4b: C = 2×A + 0 which makes C = 2×A
Hint #8
Substitute 2×A for C, and 0 for E in eq.5a: F = A + 2×2×A - 0 which becomes F = A + 4×A which makes F = 5×A
Hint #9
Substitute 2×A for C, 5×A for F, and 0 for E in eq.3: A + 2×A + 5×A = B + 0 which makes 8×A = B and which also makes D = B = 8×A
Solution
Substitute 8×A for B and D, 2×A for C, 0 for E, and 5×A for F in eq.1: A + 8×A + 2×A + 8×A + 0 + 5×A = 24 which simplifies to 24×A = 24 Divide both sides by 24: 24×A ÷ 24 = 24 ÷ 24 which means A = 1 making B = D = 8×A = 8 × 1 = 8 C = 2×A = 2 × 1 = 2 F = 5×A = 5 × 1 = 5 and ABCDEF = 182805