Puzzle for August 21, 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.
* AB and EF are 2-digit numbers (not A×B or E×F).
Scratchpad
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Hint #1
In eq.4, replace B with C + D + E (from eq.3): C + D + E + C = A + D + E – F which becomes 2×C + D + E = A + D + E – F Subtract D and E from each side of the equation above: 2×C + D + E – D – E = A + D + E – F – D – E which simplifies to eq.4a) 2×C = A – F
Hint #2
In eq.5, add C to both sides, and subtract E from both sides: C + E + C – E = A – C + C – E which becomes eq.5a) 2×C = A – E In eq.5a, replace 2×C with A – F (from eq.4a): A – F = A – E Subtract A from both sides of the equation above: A – F – A = A – E – A which becomes –F = –E Since E and F must be non-negative integers, the above equation makes: F = E
Hint #3
In eq.2, add D to both sides, and subtract F from both sides: A – D + D – F = C + F + D – F which becomes A – F = C + D In the equation above, substitute 2×C for A – F (from eq.4a): 2×C = C + D Subtract C from both sides: 2×C – C = C + D – C which makes C = D
Hint #4
Substitute C for D, and E for F in eq.4: B + C = A + C + E – E which becomes B + C = A + C Subtract C from each side of the above equation: B + C – C = A + C – C which makes B = A
Hint #5
eq.6 may be written as: D + 10×E + F = 10×A + B Substitute E for F, and C + D + E for B (from eq.3) in the equation above: D + 10×E + E = 10×A + C + D + E Subtract D, E, and 10×A from both sides: D + 10×E + E – 10×A – D – E = 10×A + C + D + E – 10×A – D – E which simplifies to eq.6a) 10×E – 10×A = C
Hint #6
Substitute (10×E – 10×A) for C (from eq.6a) in eq.5a: 2×(10×E – 10×A) = A – E which becomes 20×E – 20×A = A – E Add E and 20×A to each side of the equation above: 20×E – 20×A + E + 20×A = A – E + E + 20×A which becomes 21×E = 21×A Divide both sides by 21: 21×E ÷ 21 = 21×A ÷ 21 which makes E = A and also makes F = E = A = B
Hint #7
Substitute A for F in eq.4a: 2×C = A – A which makes 2×C = 0 which means C = 0 and also means D = C = 0
Solution
Substitute A for B and E and F, and 0 for C and D in eq.1: A + A + 0 + 0 + A + A = 28 which simplifies to 4×A = 28 Divide both sides of the equation above by 4: 4×A ÷ 4 = 28 ÷ 4 which means A = 7 making A = B = E = F = 7 and ABCDEF = 770077