# Cohomology Operations: Lectures by N.E. Steenrod. by N. E. Steenrod, David B. A. Epstein By N. E. Steenrod, David B. A. Epstein

Written and revised through D. B. A. Epstein.

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Wha t i s the larges t number m fo r whic h i t i s tru e tha t " a numbe r wit h m digit s tha t i s relatively prim e t o 2005604901 3 0 i s prime" ? 1090/stml/045/06 Simultaneous Congruences Theorem. Let a and b be relatively prime nonzero numbers. For any given numbers m and n, the simultaneous congruences x = m mo d a and x = n mo d b are equivalent to a single congruence x = k mo d ab for some k. Proof. Th e assumptio n tha t a and b are relatively prim e mean s tha t their greates t commo n diviso r i s 1 , s o th e augmente d Euclidea n al gorithm find s solution s o f 1 -f ub = va an d 1 + xa = yb.

4. Doubl e Congruence s an d th e Euclidea n Algorith1 m 9 When a > 6 , a n analogou s argumen t applie s wit h th e role s o f a and b reversed. Thus, congruenc e mo d [a , b] i s the same as congruence mo d [a , b — a] when a < b and th e sam e a s congruenc e mo d [a — b, b] whe n a > b. Repetition o f thi s proces s until 2 th e tw o number s ar e th e sam e (i n the abov e exampl e the y bot h becom e 3 ) prove s th e theorem , becaus e it show s tha t congruenc e mo d [a , b] is th e sam e a s congruenc e mo d [c, c] fo r som e c , an d i t i s clea r fro m th e definitio n tha t congruenc e mod [c , c] is th e sam e a s congruenc e mo d c .

When a and c are relatively prime nonzero numbers, the order of a mod c is the smallest solution b of a b = 1 mod c. The propositio n state s tha t th e solution s b > 0 o f a b = 1 are th e nonzer o multiple s o f th e orde r o f a mod c . mod c 40 Higher Arithmeti c Problem. Given relatively prime nonzero numbers a and c, find the order of a mod c . The proo f o f th e propositio n show s tha t th e orde r o f a mod c i s at mos t c , s o th e orde r ca n b e foun d b y computin g a b mo d c fo r al l numbers b < c.