Puzzle for January 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 the left and right sides of eq.6 to the left and right sides of eq.4, respectively: B + E + A - B = F - E + B - F which becomes E + A = -E + B Add E to each side of the above equation: E + A + E = -E + B + E which becomes eq.4a) 2×E + A = B
Hint #2
In eq.5, replace B with 2×E + A (from eq.4a): 2×E + A - A = A - E which becomes 2×E = A - E Add E to each side of the above equation: 2×E + E = A - E + E which makes 3×E = A
Hint #3
In eq.4a, substitute 3×E for A: 2×E + 3×E = B which makes 5×E = B
Hint #4
Substitute 3×E for A, and 5×E for B in eq.2: C = 3×E + 5×E which means C = 8×E
Hint #5
In eq.4, replace B with 5×E: 5×E + E = F - E Add E to each side of the above equation: 5×E + E + E = F - E + E which means 7×E = F
Hint #6
Substitute 7×E for F in eq.3: D = E + 7×E which means D = 8×E
Solution
Substitute 3×E for A, 5×E for B, 8×E for C and D, and 7×E for F in eq.1: 3×E + 5×E + 8×E + 8×E + E + 7×E = 32 which simplifies to 32×E = 32 Divide both sides by 32: 32×E ÷ 32 = 32 ÷ 32 which means E = 1 making A = 3×E = 3 × 1 = 3 B = 5×E = 5 × 1 = 5 C = D = 8×E = 8 × 1 = 8 F = 7×E = 7 × 1 = 7 and ABCDEF = 358817