## Alphametic puzzles — Part 2

(1)

$(x, \; y, \; z)$   are positive integers in arithmetic sequence such that

$x \; < \; y \; < \; z$,

$z^3 \; - \; y^3 \; = \; TWO$     and     $y^3 \; - \; x^3 \; = \; TOW$

Find   $(x, \; y, \; z)$

(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:

$one \; \equiv \; 1 \; \mod{ \,two \,}$
$one \; \equiv \; 1 \; \mod{ \,six \,}$
$six \; \equiv \; 0 \; \mod{ \,two \,}$

(3)

$SWAN \; + \; MEDIC \; = \; IDIOM$

Solve in base-10

math grad - Interest: Number theory
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### 6 Responses to Alphametic puzzles — Part 2

1. pipo says:

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

• benvitalis says:

$865 \; \equiv \; 1 \; \mod{ \, 108 \,}$
$865 \; \equiv \; 1 \; \mod{ \, 432 \,}$
$432 \; \equiv \; 0 \; \mod{ \, 108 \,}$

2. pipo says:

Oops you are right

3. David @InfinitelyManic says:

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

4. pipo says:

And if you want to know how she ACTS like two CATS then:
7 12 17: 1385 and 3185
ACTS = 1385
CATS = 3185

5. David @InfinitelyManic says:

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