Puzzle for August 30, 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
Add both D and E to each side of eq.4: B – D + E + D + E = A – E + D + E which becomes eq.4a) B + 2×E = A + D Add both E and F to each side of eq.5: D – E + E + F = B + E – F + E + F which becomes eq.5a) D + F = B + 2×E
Hint #2
In eq.4a, replace B + 2×E with D + F (from eq.5a): D + F = A + D Subtract D from both sides of the above equation: D + F – D = A + D – D which means F = A
Hint #3
In eq.2, replace F with A: C + A = A + B Subtract A from both sides of the above equation: C + A – A = A + B – A which means C = B
Hint #4
eq.6 may be written as: 10×A + B + D + E = 10×C + D In the equation above, substitute B for C: 10×A + B + D + E = 10×B + D Subtract 10×A, B, and D from both sides: 10×A + B + D + E – 10×A – B – D = 10×B + D – 10×A – B – D which simplifies to eq.6a) E = 9×B – 10×A
Hint #5
Substitute A for F, B for C, and 9×B – 10×A for E (from eq.6a) in eq.3: A = B – (9×B – 10×A) which is equivalent to A = B – 9×B + 10×A Subtract 10×A from both sides of the above equation: A – 10×A = B – 9×B + 10×A – 10×A which becomes –9×A = –8×B Divide both sides by (–8): –9×A ÷ (–8) = –8×B ÷ (–8) which makes eq.3a) 1⅛×A = B
Hint #6
Substitute (1⅛×A) for B in eq.6a: E = 9×(1⅛×A) – 10×A which becomes E = 10⅛×A – 10×A which means E = ⅛×A Multiply both sides by 8: 8 × E = 8 × ⅛×A which makes 8×E = A which also makes F = A = 8×E
Hint #7
Substitute (8×E) for A in eq.3a: 1⅛×(8×E) = B which makes 9×E = B and which makes C = B = 9×E
Hint #8
Substitute 9×E for B, and 8×E for A in eq.4a: 9×E + 2×E = 8×E + D Subtract 8×E from both sides of the above equation: 9×E + 2×E – 8×E = 8×E + D – 8×E which makes 3×E = D
Solution
Substitute 8×E for A and F, 9×E for B and C, and 3×E for D in eq.1: 8×E + 9×E + 9×E + 3×E + E + 8×E = 38 which simplifies to 38×E = 38 Divide both sides by 38: 38×E ÷ 38 = 38 ÷ 38 which means E = 1 making A = F = 8×E = 8 × 1 = 8 B = C = 9×E = 9 × 1 = 9 D = 3×E = 3 × 1 = 3 and ABCDEF = 899318