Puzzle for October 15, 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.
Scratchpad
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Hint #1
Add B to both sides of eq.5: A – B + B = D + F + B which becomes A = B + D + F In eq.4, replace A with B + D + F: B + C + D = B + D + F + F Subtract B and D from both sides of the above equation: B + C + D – B – D = B + D + F + F – B – D which simplifies to eq.4a) C = 2×F
Hint #2
Add E to both sides of eq.6: E + F + E = C – E + E which becomes 2×E + F = C In eq.2, replace C with 2×E + F: 2×E + F = B + E + F Subtract E and F from both sides of the above equation: 2×E + F – E – F = B + E + F – E – F which makes E = B
Hint #3
In eq.2, substitute 2×F for C, and B for E: 2×F = B + B + F which becomes 2×F = 2×B + F Subtract F from each side of the equation above: 2×F – F = 2×B + F – F which makes F = 2×B
Hint #4
Substitute (2×B) for F in eq.4a: C = 2×(2×B) which makes C = 4×B
Hint #5
Substitute 4×B for C, and B for E in eq.3: D = 4×B + B which makes D = 5×B
Hint #6
Substitute 5×B for D, and 2×B for F in eq.5: A – B = 5×B + 2×B which becomes A – B = 7×B Add B to both sides of the above equation: A – B + B = 7×B + B which makes A = 8×B
Solution
Substitute 8×B for A, 4×B for C, 5×B for D, B for E, and 2×B for F into eq.1: 8×B + B + 4×B + 5×B + B + 2×B = 21 which simplifies to 21×B = 21 Divide both sides of the equation above by 21: 21×B ÷ 21 = 21 ÷ 21 which means B = 1 making A = 8×B = 8×1 = 8 C = 4×B = 4×1 = 4 D = 5×B = 5×1 = 5 E = B = 1 F = 2×B = 2×1 = 2 and ABCDEF = 814512