Puzzle for January 27, 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.
* AB is a 2-digit number (not A×B).
Scratchpad
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Hint #1
Add B and D to both sides of eq.4: F – B + D + B + D = A + B – D + B + D which becomes eq.4a) F + 2×D = A + 2×B eq.6 may be written as: 10×A + B – D = E + F + D – B Add B and D to both sides of the above equation: 10×A + B – D + B + D = E + F + D – B + B + D which becomes eq.6a) 10×A + 2×B = E + F + 2×D
Hint #2
In eq.6a, replace F + 2×D with A + 2×B (from eq.4a): 10×A + 2×B = E + A + 2×B Subtract A and 2×B from both sides of the equation above: 10×A + 2×B – A – 2×B = E + A + 2×B – A – 2×B which simplifies to 9×A = E
Hint #3
Add D to both sides of eq.2: E – D + D = F + D + D which becomes eq.2a) E = F + 2×D In eq.2a, replace E with 9×A, and F + 2×D with A + 2×B (from eq.4a): 9×A = A + 2×B Subtract A from each side of the equation above: 9×A – A = A + 2×B – A which makes 8×A = 2×B Divide both sides by 2: 8×A ÷ 2 = 2×B ÷ 2 which makes 4×A = B
Hint #4
Add C and D to both sides of eq.3: C – D + C + D = B – C + D + C + D which becomes 2×C = B + 2×D In the above equation, substitute 4×A for B: 2×C = 4×A + 2×D Divide both sides by 2: 2×C ÷ 2 = (4×A + 2×D) ÷ 2 which becomes eq.3a) C = 2×A + D
Hint #5
Substitute 9×A for E in eq.2a: 9×A = F + 2×D Subtract 2×D from each side of the above equation: 9×A – 2×D = F + 2×D – 2×D which becomes eq.2b) 9×A – 2×D = F
Hint #6
Substitute 9×A – 2×D for F (from eq.2b), 9×A for E, and 2×A + D for C (from eq.3a) in eq.5: 9×A – 2×D + A – D = 9×A – A + 2×A + D + D which becomes 10×A – 3×D = 10×A + 2×D In the equation above, subtract 10×A from both sides, and add 3×D to both sides: 10×A – 3×D – 10×A + 3×D = 10×A + 2×D – 10×A + 3×D which makes 0 = 5×D which means 0 = D
Hint #7
Substitute 0 for D in eq.2b: 9×A – 2×0 = F which makes 9×A = F
Hint #8
Substitute 0 for D in eq.3a: 2×A + 0 = C which makes 2×A = C
Solution
Substitute 4×A for B, 2×A for C, 0 for D, and 9×A for E and F in eq.1: A + 4×A + 2×A + 0 + 9×A + 9×A = 25 which simplifies to 25×A = 25 Divide both sides of the above equation by 25: 25×A ÷ 25 = 25 ÷ 25 which means A = 1 making B = 4×A = 4×1 = 4 C = 2×A = 2×1 = 2 E = F = 9×A = 9×1 = 9 and ABCDEF = 142099