i

r

W

i

i

!» }
i

 

202 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

THE CHUQUET-BAKER EGG PROBLEM (!)

487. A mayde carieng egges vnto the market, and it hap-
pened a merrye Fellowe to meete her, who began to ieste with
her in suche sorte, that he ouerthrewe her Basket, and brake
all her egges ; and the mayde beeing much displeased with him
for breaking of the same, sayde very earnestly unto him, y®
should pay for them, the man considering with himself, that
by his folly they were broken, he demaunded of her what
niiber she had. The silly poor wenche coulde not well reckon,
sayde vnto him that she could not well tell him, but sayde
she, when I did put them into my Basket by 2 and by 2, there
remayned 1 egge: and when I counted them by 3 and by 3,
there remayned 1: and when I did recken them by 4 and by
4, there remayned still 1; but when I did counte them by 5
and by 5, there remayned none. The question is to know
howe manye egges the mayde had in all?

Solution. Let x be the required number of egges.

Then by the hypothesis «x =1 (mod 2)
x = 1 (mod 3)
x =1 (mod 4)
- x =0 (mod 5)
From the first three of these statements we find by § 277
that x =1 (mod 12)
Then, since x = 1 (mod 12) and x = 0 (mod 5)
and —2X1245X5=1,
x = 25 (mod 60) § 284.
(Here the m of §284 is 12, n =35, a=x, b=1, ¢ =0,

k==21=17%)
Since x = 25 (mod 60), x — 25 = 60g and x = 60g + 25.
Since x must be positiv, evidently ¢ must be positiv or zero.
The smallest value of x is got by putting ¢ = 0 and is 25,
the next smallest is got by putting ¢ = 1 and is 85.
The number of egges was probably either 25 or 85.

(1) See Vera Sanford, l. c., p. 224.