**(1)**

are positive integers in arithmetic sequence such that

,

and

Find

(2)

Substitute each of the letters with a different decimal digit from 0 to 9, to satisfy the following system of simultaneous modular alphametic equations:

(3)

Solve in base-10

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## About benvitalis

math grad - Interest: Number theory

Puzzle one:

(x, y, z) = (8, 9, 10)

Puzzle two:

581 ≡ 1 mod 145

581 ≡ 1 mod 290

290 ≡ 0 mod 145

Puzzle three has many solutions, like:

2437

59168+

61605

Oops you are right

I see Pipo already posted a solution to #1

8 9 10 271 217

However if you want to know how to ACT like a CAT then:

2 9 10 271 721

ACT = 271

CAT = 721

And if you want to know how she ACTS like two CATS then:

7 12 17: 1385 and 3185

ACTS = 1385

CATS = 3185

When PUPILS SLIPUP and hate math. I did not check for conditions in 2).

100^3 -73^3 = 610983 = PUPILS

73^3 -1^3 = 389016 = SLIPUP