130 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

APPLICATION OF CONGRUENCES
TO CHECKING ARITHMETICAL WORK

310. One of the most interesting applications of congru-
ences is to checking arithmetical work.

For example, suppose we are required to add 374 and 625
and to check the work. If we choose the modulus 6, we find
374 = 2 and 625 = 1. Hence 374 + 625 must be congruent
to 2 + 1 (§206). Adding 374 and 625 we get the sum 999.
By § 202 we find that 999 is congruent to 3. Thus the work
checks.

We may indicate the work as follows:

374 = 2 (mod 6)

625 = 1 (mod 6)
999 = 3 (mod 6)

As other examples we may take the following:

437 = 5 (mod 6) 359 = 2 (mod 7)
541 =1 i 263 =4 s
978 = 6 = 0 (mod 6) e
897 =8 =1 (mod 7)
In subtraction:
3460 = 4 (mod 9) 87653 = 2 (mod 3)
2159 =8 e 98421 = 0 .
1301 = — 4 =5 (mod 9) — 10768 = 2 (mod 3)
In multiplication:
67 = 2 (mod 5) 37= 2 (mod>5)
20=14 > 47 = -3 >
603 259
134 148

1943 = 8 = 3 (mod 5) 1739 = — 6= — 1 =4 (mod 5)