Puzzle for October 24, 2020 ( )
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.
* CD, DE, and EF are 2-digit numbers (not C×D, D×E, or E×F).
Scratchpad
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Hint #1
In eq.2, replace A with C + E (from eq.5): eq.2a) D = C + E – F
Hint #2
In eq.3, add B and C to both sides, and subtract F from each side: B – C + F + B + C – F = E – B + B + C – F which simplifies to 2×B = E + C – F which may be written as 2×B = C + E – F In eq.2a, replace C + E – F with 2×B: D = 2×B
Hint #3
eq.6 may be written as: C + 10×D + E = 10×C + D + 10×E + F Subtract C, D, and E from each side of the equation above: C + 10×D + E – C – D – E = 10×C + D + 10×E + F – C – D – E which simplifies to eq.6a) 9×D = 9×C + 9×E + F
Hint #4
In eq.2a, add F to each side, and subtract D from each side: D + F – D = C + E – F + F – D which becomes F = C + E – D In eq.6a, substitute C + E – D for F: 9×D = 9×C + 9×E + C + E – D which becomes 9×D = 10×C + 10×E – D Add D to both sides of the above equation: 9×D + D = 9×C + 9×E + C + E – D + D which becomes eq.6b) 10×D = 10×C + 10×E
Hint #5
Divide both sides of eq.6b by 10: 10×D ÷ 10 = (10×C + 10×E) ÷ 10 which becomes D = C + E Substitute 2×B for D, and A for C + E (from eq.5) in the equation above: 2×B = A which means D = A = 2×B
Hint #6
Substitute A for D in eq.2: A = A – F Subtract A from both sides of the above equation: A – A = A – F – A which makes 0 = –F which means 0 = F
Hint #7
Substitute 2×B for A in eq.5: 2×B = C + E Subtract E from each sides of the equation above: 2×B – E = C + E – E which becomes eq.5a) 2×B – E = C
Hint #8
Substitute 2×B for A in eq.4: E – C = 2×B + B + C – E which becomes E – C = 3×B + C – E Add C and E to both sides of the equation above: E – C + C + E = 3×B + C – E + C + E which becomes eq.4a) 2×E = 3×B + 2×C
Hint #9
Substitute (2×B – E) for C (from eq.5a) in eq.4a: 2×E = 3×B + 2×(2×B – E) which is equivalent to 2×E = 3×B + 4×B – 2×E which becomes 2×E = 7×B – 2×E Add 2×E to each side of the equation above: 2×E + 2×E = 7×B – 2×E + 2×E which makes 4×E = 7×B Divide both sides by 4: 4×E ÷ 4 = 7×B ÷ 4 which makes E = 1¾×B
Hint #10
Substitute 1¾×B for E in eq.5a: 2×B – 1¾×B = C which makes ¼×B = C
Solution
Substitute 2×B for A and D, ¼×B for C, 1¾×B for E, and 0 for F in the equation above: 2×B + B + ¼×B + 2×B + 1¾×B + 0 = 28 which simplifies to 7×B = 28 Divide both sides of the equation above by 7: 7×B ÷ 7 = 28 ÷ 7 which means B = 4 making A = D = 2×B = 2 × 4 = 8 C = ¼×B = ¼ × 4 = 1 E = 1¾×B = 1¾ × 4 = 7 and ABCDEF = 841870