Puzzle for March 8, 2022 ( )
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.
Scratchpad
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Hint #1
In eq.4, replace F with D + E (from eq.2): D – E = D + E – D which becomes D – E = E Add E to both sides of the above equation: D – E + E = E + E which makes D = 2×E
Hint #2
In eq.2, replace D with 2×E: F = 2×E + E which makes F = 3×E
Hint #3
In eq.6, substitute 3×E for F, and 2×E for D in eq.6: E + 3×E = A + 2×E – E which becomes 4×E = A + E Subtract E from each side of the above equation: 4×E – E = A + E – E which makes 3×E = A
Hint #4
Substitute 3×E for A in eq.3: 3×E = B + C Subtract C from each side of the equation above: 3×E – C = B + C – C which becomes eq.3a) 3×E – C = B
Hint #5
Substitute 2×E for D, and 3×E – C for B (from eq.3a) in eq.5: C + 2×E = 3×E – C – C + E which becomes C + 2×E = 4×E – 2×C In the equation above, subtract 2×E from both sides, and add 2×C to both sides: C + 2×E – 2×E + 2×C = 4×E – 2×C – 2×E + 2×C which makes 3×C = 2×E Divide both sides of the above equation by 3: 3×C ÷ 3 = 2×E ÷ 3 which makes C = ⅔×E
Hint #6
Substitute ⅔×E for C in eq.3a: 3×E – ⅔×E = B which makes 2⅓×E = B
Solution
Substitute 3×E for A and F, 2⅓×E for B, ⅔×E for C, and 2×E for D in eq.1: 3×E + 2⅓×E + ⅔×E + 2×E + E + 3×E = 36 which simplifies to 12×E = 36 Divide both sides of the above equation by 12: 12×E ÷ 12 = 36 ÷ 12 which means E = 3 making A = F = 3×E = 3 × 3 = 9 B = 2⅓×E = 2⅓ × 3 = 7 C = ⅔×E = ⅔ × 3 = 2 D = 2×E = 2 × 3 = 6 and ABCDEF = 972639