Puzzle for March 9, 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.
* AB and BC are 2-digit numbers (not A×B or B×C). CDE and DEF are 3-digit numbers (not C×D×E or D×E×F).
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Hint #1
eq.6 may be written as: 10×B + C + 100×D + 10×E + F = 100×C + 10×D + E In the equation above, substitute (D - E) for C (from eq.4): 10×B + (D - E) + 100×D + 10×E + F = 100×(D - E) + 10×D + E which becomes 10×B + 101×D + 9×E + F = 100×D - 100×E + 10×D + E which becomes 10×B + 101×D + 9×E + F = 110×D - 99×E Add (99×E - 101×D) to each side: 10×B + 101×D + 9×E + F + (99×E - 101×D) = 110×D - 99×E + (99×E - 101×D) which simplifies to eq.6a) 10×B + 108×E + F = 9×D
Hint #2
Since D must be a one-digit non-negative integer, then: D ≤ 9 which makes 9×D ≤ 81 Replace 9×D with 10×B + 108×E + F (from eq.6a) in the inequality above: eq.6b) 10×B + 108×E + F ≤ 81 Since B, E, and F are all one-digit non-negative integers, then if E ≥ 1: 108×E ≥ 108 which would make 10×B + 108×E + F ≥ 108 However, the above inequality contradicts eq.6b, which means E not ≥ 1 which makes E = 0
Hint #3
In eq.4, replace E with 0: C = D - 0 which makes C = D
Hint #4
Subtract the left and right sides of eq.3 from the left and right sides of eq.2, respectively: D + E - (D - F) = A + B - (B - A) which becomes D + E - D + F = A + B - B + A which simplifies to E + F = 2×A Substitute 0 for E in the above equation: 0 + F = 2×A which makes F = 2×A
Hint #5
Substitute 0 for E in eq.2: D + 0 = A + B which makes eq.2a) D = A + B
Hint #6
eq.5 may be written as: 10×A + B = A + C + D In the above equation, substitute D for C, and subtract A from each side: 10×A + B - A = A + D + D - A which becomes 9×A + B = 2×D Substitute (A + B) for D (from eq.2a): 9×A + B = 2×(A + B) which becomes 9×A + B = 2×A + 2×B Subtract both 2×A and B from each side: 9×A + B - 2×A - B = 2×A + 2×B - 2×A - B which simplifies to 7×A = B
Hint #7
Substitute 7×A for B in 2×A: D = A + 7×A which makes D = 8×A
Solution
Substitute 7×A for B, 8×A for C and D, 0 for E, and 2×A for F in eq.1: A + 7×A + 8×A + 8×A + 0 + 2×A = 26 which simplifies to 26×A = 26 Divide both sides of the above equation by 26: 26×A ÷ 26 = 26 ÷ 26 which makes A = 1 making B = 7×A = 7 × 1 = 7 C = D = 8×A = 8 × 1 = 8 F = 2×A = 2 × 1 = 2 and ABCDEF = 178802