Puzzle for July 19, 2021 ( )
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 F to both sides of eq.4: A – F + F = C + F + F which becomes A = C + 2×F In eq.3, replace A with C + 2×F: C + E = C + 2×F + F which becomes C + E = C + 3×F Subtract C from both sides of the above equation: C + E – C = C + 3×F – C which makes E = 3×F
Hint #2
In eq.6, replace E with 3×F: 3×F + F = B – F which becomes 4×F = B – F Add F to both sides of the above equation: 4×F + F = B – F + F which makes 5×F = B
Hint #3
In eq.5, substitute 5×F for B: 5×F + F = C – F which becomes 6×F = C – F Add F to both sides of the equation above: 6×F + F = C – F + F which becomes 7×F = C
Hint #4
Substitute 7×F for C in eq.4: A – F = 7×F + F which becomes A – F = 8×F Add F to both sides of the above equation: A – F + F = 8×F + F which makes A = 9×F
Hint #5
Substitute 3×F for E, 5×F for B, and 7×F for C in eq.2: D + 3×F = 5×F + 7×F which becomes D + 3×F = 12×F Subtract 3×F from each side of the above equation: D + 3×F – 3×F = 12×F – 3×F which makes D = 9×F
Solution
Substitute 9×F for A and D, 5×F for B, 7×F for C, and 3×F for E in eq.1: 9×F + 5×F + 7×F + 9×F + 3×F + F = 34 which simplifies to 34×F = 34 Divide both sides of the above equation by 34: 34×F ÷ 34 = 34 ÷ 34 which means F = 1 making A = D = 9×F = 9 × 1 = 9 B = 5×F = 5 × 1 = 5 C = 7×F = 7 × 1 = 7 E = 3×F = 3 × 1 = 3 and ABCDEF = 957931