By W. W. Sawyer

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176/ D 455: 28 1 Number Concepts, Prime Numbers, and the Division Algorithm Since 7j455, the original n is divisible by 7. ) B. 400 C 10a C 6/ D 211: Since 211 is not divisible by 13, there are no values of a for which the given n of Part B is divisible by 13. ) we’ll now prove Test 7/13 only for the case of a 6 digit number (so ` D 5). n/ is divisible by 13. Proof. 4 Tests for Divisibility in Base Ten 29 where the last equality follows from Eq. 7). Thus, from the definition of the sequential triple sum and Eq.

What is the number of the duodecimal representative? 12/ C 0 Hence n D 275 D 1 122 C 10 12 C 11 120 D 1TE. 38 1 Number Concepts, Prime Numbers, and the Division Algorithm Example 28. If n is 160 in base 10, what is it in base 12? 12/. ak ak 1 : : : a0 /12 ; its decimal representation can be obtained by computing n D 12k ak C 12k 1 ak 1 C : : : C 121 a1 C a0 : Example 29. Let n D 11412 . Then n D 1 122 C1 121 C4 D 144C12C4 D 16010 . Example 30. 3T 7/12 . Then n Š 3 122 C10 121 C7 120 D 432C120C7 D 559.

123; 36/ and show that it is a linear combination of 123 and 36. Solution. 123; 36/ D 3. However, now look at what the chain of substitutions from Eqs. (A), (B) and (C) gives! 2 GCD, the Euclidean Algorithm and Its Byproducts 55 linear combination” result which appears is the Euclidean Algorithm Byproduct, Theorem 8. 123; 36/ D 3 D 5b C . a; b/ is a linear combination of a and b. // In number theory, the process of Example 40 is (sometimes) called BACKWARDS Look at the pattern in Eqs. (A), (B), (C), and (D)—the three 15’s run diagonally, then the three 6’s, then the two 3’s and note that 15, 6, and 3 are remainders.

### A Concrete Approach to Abstract Algebra by W. W. Sawyer

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