The Egyptian pharaoh (part 3)

Number Theory Level 3

ABCCCCCCCDE × 4 =

EDCCCCCCCBA

Hardly had the Egyptian pharaoh been happy with the users of Brilliant for interpreting his dream, when he saw the above multiplication. Could you interpret it for him, show him your brilliance and tell him what the eleven digit number ABCCCCCCCDE is??


The answer is 21999999978.

Mohd Sasa by Mohd Sasa , 49, Egypt

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1 solution

James Moors
May 2, 2015

The leftmost digits give us: 4 A ≤ E < 10 ⇒ A ∈ { 1 , 2 } ; the rightmost: 4 E = A ( mod 10 ) ⇒ A = 2 ⇒ E = 8 ( since A is even, E ≥ 4 A ) . \text{The leftmost digits give us: }4A\leq E<10 \Rightarrow A \in \{1,2\};\\ \text{the rightmost: }4E = A \:( \text{mod } 10) \Rightarrow A = 2 \Rightarrow E = 8 \\ (\text{since }A \text{ is even, }E \geq 4A). Similarly, for B B and D D , we have: 4 B ≤ D < 10 ⇒ B ∈ { 1 , 2 } ; and, since 4 E = 30 + 2 , 4 D + 3 = B ( mod 10 ) ⇒ B = 1 ⇒ D = 7. 4B \leq D < 10 \Rightarrow B \in \{1,2\};\\ \text{and, since } 4E = 30+2, 4D+3 = B (\text{mod } 10) \Rightarrow B = 1 \Rightarrow D = 7. Finally, note that the carry from 4 D 4D must be the same as for 4 C 4C : 4 C + 3 = 30 + C 3 C = 27 C = 9 4C + 3 = 30 + C\\ 3C = 27\\ C = 9 Giving the answer of 21999999978 \boxed{21999999978}

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