Puzzle for April 18, 2022 ( )
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
In eq.6, replace F with D + E (from eq.4): B + D = D + E – B In the above equation, subtract D from both sides, and add B to both sides: B + D – D + B = D + E – B – D + B which simplifies to 2×B = E
Hint #2
In eq.5, replace E with 2×B: B + 2×B = A – 2×B which becomes 3×B = A – 2×B Add 2×B to both sides of the above equation: 3×B + 2×B = A – 2×B + 2×B which makes 5×B = A
Hint #3
In eq.3, substitute 5×B for A: D = 5×B + B which makes D = 6×B
Hint #4
Substitute 6×B for D in eq.2: C = B + 6×B which makes C = 7×B
Hint #5
Substitute 6×B for D in eq.6: B + 6×B = F – B which becomes 7×B = F – B Add B to both sides of the equation above: 7×B + B = F – B + B which makes 8×B = F
Solution
Substitute 5×B for A, 7×B for C, 6×B for D, 2×B for E, and 8×B for F in eq.1: 5×B + B + 7×B + 6×B + 2×B + 8×B = 29 which simplifies to 29×B = 29 Divide both sides of the above equation by 29: 29×B ÷ 29 = 29 ÷ 29 which means B = 1 making A = 5×B = 5 × 1 = 5 C = 7×B = 7 × 1 = 7 D = 6×B = 6 × 1 = 6 E = 2×B = 2 × 1 = 2 F = 8×B = 8 × 1 = 8 and ABCDEF = 517628