Puzzle for April 26, 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.5, replace D with B + E (from eq.4): B + B + E = E + F which becomes 2×B + E = E + F Subtract E from each side of the equation above: 2×B + E – E = E + F – E which makes 2×B = F
Hint #2
In eq.2, replace D with B + E (from eq.4), and replace F with 2×B: A = B + E + 2×B which becomes eq.2a) A = 3×B + E
Hint #3
In eq.6, substitute 3×B + E for A (from eq.2a), and 2×B for F: C + E = 3×B + E + 2×B which becomes C + E = 5×B + E Subtract E from both sides of the equation above: C + E – E = 5×B + E – E which makes C = 5×B
Hint #4
Substitute 5×B for C in eq.3: E = B + 5×B which makes E = 6×B
Hint #5
Substitute 6×B for E in eq.4: D = B + 6×B which makes D = 7×B
Hint #6
Substitute 6×B for E in eq.2a: A = 3×B + 6×B which makes A = 9×B
Solution
Substitute 9×B for A, 5×B for C, 7×B for D, 6×B for E, and 2×B for F in eq.1: 9×B + B + 5×B + 7×B + 6×B + 2×B = 30 which simplifies to 30×B = 30 Divide both sides of the equation above by 30: 30×B ÷ 30 = 30 ÷ 30 which means B = 1 making A = 9×B = 9 × 1 = 9 C = 5×B = 5 × 1 = 5 D = 7×B = 7 × 1 = 7 E = 6×B = 6 × 1 = 6 F = 2×B = 2 × 1 = 2 and ABCDEF = 915762