Puzzle for June 22, 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.
* CD and EF are 2-digit numbers (not C×D or E×F).
Scratchpad
Help Area
Hint #1
In eq.3, replace D with A + B (from eq.2): B + A + B + E = A + F Subtract A from each side of the above equation: B + A + B + E - A = A + F - A eq.3a) 2×B + E = F
Hint #2
In eq.4, replace F with 2×B + E (from eq.3a): C - B = 2×B + E - E which becomes C - B = 2×B Add B to both sides: C - B + B = 2×B + B which makes C = 3×B
Hint #3
eq.6 may be written as: A - B + 10×C + D + E = 10×E + F Subtract E from each side of the above equation: A - B + 10×C + D + E - E = 10×E + F - E which becomes A - B + 10×C + D = 9×E + F which also may be written as eq.6a) A - B + 10×C + D = 8×E + E + F
Hint #4
In eq.6a, substitute C + D for E + F (from eq.5): A - B + 10×C + D = 8×E + C + D Subtract both C and D from each side of the equation above: A - B + 10×C + D - C - D = 8×E + C + D - C - D which becomes eq.6b) A - B + 9×C = 8×E
Hint #5
Subtract both B and F from each side of eq.3: B + D + E - B - F = A + F - B - F which becomes D + E - F = A - B Substitute D + E - F for A - B in eq.6b: D + E - F + 9×C = 8×E Subtract E from each side: D + E - F + 9×C - E = 8×E - E which becomes eq.6c) D - F + 9×C = 7×E
Hint #6
Subtract both C and F from each side of eq.5: E + F - C - F = C + D - C - F which becomes E - C = D - F Substitute E - C for D - F in eq.6c: E - C + 9×C = 7×E Subtract E from each side: E - C + 9×C - E = 7×E - E which simplifies to 8×C = 6×E Substitute 3×B for C: 8×3×B = 6×E which becomes 24×B = 6×E Divide both sides by 6: 24×B ÷ 6 = 6×E ÷ 6 which makes 4×B = E
Hint #7
Substitute 3×B for C, and 4×B for E in eq.4: 3×B - B = F - 4×B Add 4×B to both sides: 3×B - B + 4×B = F - 4×B + 4×B which makes 6×B = F
Hint #8
Substitute 4×B for E, 6×B for F, and 3×B for C in eq.5: 4×B + 6×B = 3×B + D Subtract 3×B from both sides: 4×B + 6×B - 3×B = 3×B + D - 3×B which makes 7×B = D
Hint #9
Substitute 7×B for D in eq.2: 7×B = A + B Subtract B from each side: 7×B - B = A + B - B which makes 6×B = A
Solution
Substitute 6×B for A and F, 3×B for C, 7×B for D, and 4×B for E in eq.1: 6×B + B + 3×B + 7×B + 4×B + 6×B = 27 which becomes 27×B = 27 Divide both sides by 27: 27×B ÷ 27 = 27 ÷ 27 B = 1 making A = F = 6×B = 6×1 = 6 C = 3×B = 3×1 = 3 D = 7×B = 7×1 = 7 E = 4×B = 4×1 = 4 and ABCDEF = 613746