Puzzle for May 11, 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.
* DE is a 2-digit number (not D×E).
Scratchpad
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Hint #1
In eq.6, add F to both sides, and subtract B from both sides: B + E + F - B = A - F + F - B which becomes E + F = A - B In eq.2, replace E + F with A - B, and replace C + D with A + B (from eq.4): A - B = B + A + B which becomes A - B = A + 2×B In the above equation, add B to both sides, and subtract A from both sides: A - B + B - A = A + 2×B + B - A which simplifies to 0 = 3×B which means 0 = B
Hint #2
Subtract both B and C from each side of eq.4: A + B - B - C = C + D - B - C which becomes A - C = D - B In eq.3, replace A - C with D - B: eq.3a) C - E = D - B
Hint #3
Add B and E to both sides of eq.3a: C - E + B + E = D - B + B + E which becomes C + B = D + E which may be written as B + C = D + E In eq.2, substitute D + E for B + C: E + F = D + E + D Subtract E from both sides of the above equation: E + F - E = D + E + D - E which means F = 2×D
Hint #4
Substitute 2×D for F, and 0 for B in eq.2: E + 2×D = 0 + C + D Subtract D from both sides of the above equation: E + 2×D - D = 0 + C + D - D which becomes E + D = C which may be written as eq.2a) D + E = C
Hint #5
eq.5 may be written as: 10×D + E = A + C which may also be written as 9×D + D + E = A + C Substitute C for D + E (from eq.2a) in the above equation: 9×D + C = A + C Subtract C from both sides: 9×D + C - C = A + C - C which makes 9×D = A
Hint #6
Substitute 9×D for A, and 0 for B in eq.4: 9×D + 0 = C + D Subtract D from each side: 9×D + 0 - D = C + D - D which makes 8×D = C
Hint #7
Substitute 8×D for C in eq.2a: D + E = 8×D Subtract D from each side: D + E - D = 8×D - D which makes E = 7×D
Solution
Substitute 9×D for A, 0 for B, 8×D for C, 7×D for E, and 2×D for F in eq.1: 9×D + 0 + 8×D + D + 7×D + 2×D = 27 which becomes 27×D = 27 Divide both sides of the above equation by 27: 27×D ÷ 27 = 27 ÷ 27 which means D = 1 making A = 9×D = 9 × 1 = 9 C = 8×D = 8 × 1 = 8 E = 7×D = 7 × 1 = 7 F = 2×D = 2 × 1 = 2 and ABCDEF = 908172