Puzzle for October 3, 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 DE are 2-digit numbers (not A×B or D×E).
Scratchpad
Help Area
Hint #1
Subtract D from both sides of eq.3: D + E – D = A – D which becomes E = A – D In eq.5, replace A – D with E: C – F + E = E In the above equation, add F to both sides, and subtract E from each side: C – F + E + F – E = E + F – E which simplifies to C = F
Hint #2
In eq.2, replace C with F: A + B + F = D + E + F Subtract F from both sides of the above equation: A + B + F – F = D + E + F – F which becomes A + B = D + E Replace D + E with A (from eq.3): A + B = A Subtract A from both sides: A + B – A = A – A which means B = 0
Hint #3
eq.6 may be written as: 10×A + B – A = 10×D + E + E which becomes 9×A + B = 10×D + 2×E In the above equation, substitute (D + E) for A (from eq.3), and 0 for B: 9×(D + E) + 0 = 10×D + 2×E which is the same as 9×D + 9×E = 10×D + 2×E Subtract 2×E and 9×D from both sides: 9×D + 9×E – 2×E – 9×D = 10×D + 2×E – 2×E – 9×D which simplifies to 7×E = D
Hint #4
Substitute 7×E for D in eq.3: 7×E + E = A which makes 8×E = A
Hint #5
Substitute C for F, and 7×E for D in eq.4: C – E = 7×E – C Add both C and E to each side: C – E + C + E = 7×E – C + C + E which makes 2×C = 8×E Divide both sides by 2: 2×C ÷ 2 = 8×E ÷ 2 which makes C = 4×E and which also makes F = C = 4×E
Solution
Substitute 8×E for A, 0 for B, 4×E for C and F, and 7×E for D in eq.1: 8×E + 0 + 4×E + 7×E + E + 4×E = 24 which simplifies to 24×E = 24 Divide both sides by 24: 24×E ÷ 24 = 24 ÷ 24 which means E = 1 making A = 8×E = 8 × 1 = 8 C = F = 4×E = 4 × 1 = 4 D = 7×E = 7 × 1 = 7 and ABCDEF = 804714