Puzzle for May 8, 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 DE are 2-digit numbers (not C×D or D×E).
Scratchpad
Help Area
Hint #1
eq.6 may be written as: 10×D + E - F = C Add F to both sides of the above equation: 10×D + E - F + F = C + F which becomes eq.6a) 10×D + E = C + F
Hint #2
In eq.2, replace C + F with 10×D + E (from eq.6a): 10×D + E = D + E Subtract both D and E from the above equation: 10×D + E - D - E = D + E - D - E which simplifies to 9×D = 0 which means D = 0
Hint #3
eq.3 may be written as: A + D - E = E + F Subtract the left and right sides of the above equation from the left and right sides of eq.4, respectively: A + E - (A + D - E) = B + F - (E + F) which is equivalent to A + E - A - D + E = B + F - E - F which becomes 2×E - D = B - E Replace D with 0, and add E to both sides: 2×E - 0 + E = B - E + E which makes eq.4a) 3×E = B
Hint #4
In eq.6a, replace D with 0, and subtract C from both sides: 10×0 + E - C = C + F - C which becomes eq.6b) E - C = F eq.5 may be written as: B + E = 10×C + D + F In the above equation, substitute 3×E for B (from eq.4a), 0 for D, and E - C for F (from eq.6b): 3×E + E = 10×C + 0 + E - C which becomes 4×E = 9×C + E Subtract E from both sides: 4×E - E = 9×C + E - E which makes 3×E = 9×C Divide both sides by 3: 3×E ÷ 3 = 9×C ÷ 3 which means E = 3×C
Hint #5
Substitute 3×C for E in eq.4a: 3×3×C = B which makes 9×C = B
Hint #6
Substitute 3×C for E in eq.6b: 3×C - C = F which makes 2×C = F
Hint #7
Substitute 3×C for E, 9×C for B, and 2×C for F in eq.4: A + 3×C = 9×C + 2×C which becomes A + 3×C = 11×C Subtract 3×C from each side of the above equation: A + 3×C - 3×C = 11×C - 3×C which makes A = 8×C
Solution
Substitute 8×C for A, 9×C for B, 0 for D, 3×C for E, and 2×C for F in eq.1: 8×C + 9×C + C + 0 + 3×C + 2×C = 23 which simplifies to 23×C = 23 Divide both sides by 23: 23×C ÷ 23 = 23 ÷ 23 which makes C = 1 making A = 8×C = 8 × 1 = 8 B = 9×C = 9 × 1 = 9 E = 3×C = 3 × 1 = 3 F = 2×C = 2 × 1 = 2 and ABCDEF = 891032