I was studying modular arithmetic and began to think about the implications of two number that have the same rest of division by a number. But my conclusion fails when I apply:
Be a and b two integers numbers. If [MATH] a \equiv b \mod{c} [/MATH], then [MATH]cq_{1} - a = cq_{2} - b[/MATH] and [MATH]a - b = kc[/MATH] :
[MATH]cq_{1} = cq_{2} - b + a[/MATH][MATH]- cq_{2} + cq_{1} = - b + a[/MATH][MATH]cq_{2} - cq_{1} = b - a[/MATH]
But [MATH]b - a = -kc[/MATH], then:
[MATH]cq_{2} - cq_{1} = -kc[/MATH][MATH]q_{1} - q_{2} = k[/MATH]
With [MATH]38 \equiv 14 \mod{12} [/MATH]:
[MATH]38 - 14 = 12 * 1[/MATH], then [MATH]k = 1[/MATH]
But with [MATH]q_{1} - q_{2} = k[/MATH] :
[MATH]3 - 1 = 2[/MATH]
Source: https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence
Be a and b two integers numbers. If [MATH] a \equiv b \mod{c} [/MATH], then [MATH]cq_{1} - a = cq_{2} - b[/MATH] and [MATH]a - b = kc[/MATH] :
[MATH]cq_{1} = cq_{2} - b + a[/MATH][MATH]- cq_{2} + cq_{1} = - b + a[/MATH][MATH]cq_{2} - cq_{1} = b - a[/MATH]
But [MATH]b - a = -kc[/MATH], then:
[MATH]cq_{2} - cq_{1} = -kc[/MATH][MATH]q_{1} - q_{2} = k[/MATH]
With [MATH]38 \equiv 14 \mod{12} [/MATH]:
[MATH]38 - 14 = 12 * 1[/MATH], then [MATH]k = 1[/MATH]
But with [MATH]q_{1} - q_{2} = k[/MATH] :
[MATH]3 - 1 = 2[/MATH]
Source: https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence
