Page 56 - Numbertheory

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:( )

x = (12)(11)(22)(11)(12)(22)(11)(11)(22)(22)

= (1× 3 + 2) × 318 + (1×3 +1).316 + (2× 3 + 2) × 314

+ (1× 3 +1) × 312 + (1× 3 + 2) × 310 + (2 ×3 + 2) × 38

+ (1× 3 +1) × 36 + (1× 3 +1) × 34 + (2 × 3 + 2)× 32 + (2 ×3 + 2)

= 5× 99 + 4 ×98 + 8× 97 + 4 × 96 + 5×105

+ 8× 94 + 4× 93 + 4 × 92 + 8× 91 + 8

= 54845844889

.5

12A 3B A [Mathcounts 1986] ( )

A =0 ( ) 94 A ≠B

A =1 ( ) A =2( ) A =6()

9 12A 3B :( )
A +B +6
.9 .9

24 A +B =3 . A + B + 6 = 18 a + b + 6 = 9

. A + B = 12

3B 4 12A 3B

.B =6 B =2 .4

. A = −3 A + B = 3 B = 6

٤٣