Question #297635

Six professor begin courses on Monday, Tuesday, Wednesday, Thursday, friday, and Saturday, and announce their intention of lecturing at interval of 3,2,5,6,1and 4 days respectively the regulations of the university forbid Sunday a lecture(so that a Sunday a lecture must be omitted) when first will all six professors find themselves compelled to omit a lecture

1
Expert's answer
2022-02-16T11:38:28-0500

If the day in question is the xth (counting from and including the first monday), then x=1+2k1=2+3k2=3+4k3=4+k4=5+6k5=6+5k6=7k7\mathrm{x}=1+2 \mathrm{k}_{1}=2+3 \mathrm{k}_{2}=3+4 \mathrm{k}_{3}=4+\mathrm{k}_{4}=5+6 \mathrm{k}_{5}=6+5 \mathrm{k}_{6}=7 \mathrm{k}_{7}

where the \mathrm{k} are integers ie

 (1)x1(mod2),(2)x(mod3),(3)x3(mod4)(4)x4(mod1),(5)x5(mod6),(6)x6(mod5)(7)x0(mod7)(1) \mathrm{x} \equiv 1(\bmod 2),\\(2) \mathrm{x} \equiv(\bmod 3),\\(3) \mathrm{x} \equiv 3(\bmod 4) \\ (4) \mathrm{x} \equiv 4(\bmod 1),\\(5) \mathrm{x} \equiv 5(\bmod 6),\\(6) \mathrm{x} \equiv 6(\bmod 5) \\(7) \mathrm{x} \equiv 0(\bmod 7)

Of these congruences, (4) is no restriction and (1) and (2) are included in (3) and (5).

Of the two later, (3) shows that x\mathrm{x} is congruent to 3,7, or 11(mod12)11(\bmod 12) , and (5) that x\mathrm{x} is congruent

to 5 or 11 , so that (3) and (5) together are equivalent to x11(mod12)\mathrm{x} \equiv 11(\bmod 12) . Hence the problem is that of solving

x11(mod12),x6(mod5),x0(mod7)x \equiv 11(\bmod 12), x \equiv 6(\bmod 5), x \equiv 0(\bmod 7)

x1(mod12),x1(mod5),x0(mod7)\mathrm{x} \equiv-1(\bmod 12), \mathrm{x} \equiv 1(\bmod 5), \mathrm{x} \equiv 0(\bmod 7)

Here m1=12, m2=5, m3=7, m=420,M1=35,M2=84,M3=60\mathrm{m}_{1}=12, \mathrm{~m}_{2}=5, \mathrm{~m}_{3}=7, \mathrm{~m}=420, \mathrm{M}_{1}=35, \mathrm{M}_{2}=84, \mathrm{M}_{3}=60 .

The n\mathrm{n} are given

35n11(mod12),84n21(mod5),60n1(mod7)35 \mathrm{n}_{1} \equiv 1(\bmod 12), 84 \mathrm{n}_{2} \equiv 1(\bmod 5), 60 \mathrm{n} \equiv 1(\bmod 7)

or

n11(mod12),n21(mod5),4n31(mod7)-\mathrm{n}_{1} \equiv 1(\bmod 12),-\mathrm{n}_{2} \equiv 1(\bmod 5), 4 \mathrm{n}_{3} \equiv 1(\bmod 7)

and we can take n1=1,n2=1,n3=2\mathrm{n}_{1}=-1, \mathrm{n}_{2}=-1, \mathrm{n}_{3}=2 . Hence

 x=(1)(1)35+(1)1.84+2.0.60=49371(mod420)\mathrm{x}=(-1)(-1) 35+(-1) 1.84+2.0 .60=-49 \equiv 371(\bmod 420)

The first x\mathrm{x} satisfying the condition is 371 .


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