Question

Explain and show work:

Calculate (413,465,789 mod 6), giving an answer between 0 and 5, and using a small number of steps. Show your steps.

Answer 1

5

Step-by-step explanation:

The sum of the digits of the number is ...

(4+1+3)+(4+6+5)+(7+8+9) = 8+15+24 = 47

The sum of those digits is 4+7=11, and those digits sum to 1+1 = 2.

That is, the value of the number mod 9 (or 3) is 2.

The ones digit is odd, so the value of the number mod 2 is 1.

This combination of modulo values tells you the mod 6 result is 5.

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Additional comment

We can look at the (mod2, mod3) values of the numbers 0 to 5:

0 ⇒ (0, 0)

1 ⇒ (1, 1)

2 ⇒ (0, 2)

3 ⇒ (1, 0)

4 ⇒ (0, 1)

5 ⇒ (1, 2) . . . . the mod {2, 3} results we have for the number of interest.

This process of adding up the digits repeatedly is referred to as "casting out 9s." The result of it is the modulo 9 value of the number (with 0 mapped to 9). Checking the mod 9 result of arithmetic operations is one quick way to spot certain kinds of errors. It can also be used as part of a divisibility test for 3 or 9.