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Books Books 1 - 10 of 17 on that an integer is divisible by 9 if and only if the sum of its digits (in decimal....
" that an integer is divisible by 9 if and only if the sum of its digits (in decimal notation) is divisible by "
Problem-Solving Through Problems
by Loren C. Larson - 1992 - 352 pages
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Introduction to Analytic Number Theory

Tom M. Apostol - Mathematics - 1998 - 340 pages
...two examples to illustrate their usefulness. EXAMPLE 1 Test for divisibility by 9. An integer n > 0 is divisible by 9 if, and only if, the sum of its digits in its decimal expansion is divisible by 9. This property is easily proved using congruences. If the digits...
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Undergraduate Algebra

Serge Lang - Mathematics - 2001 - 371 pages
...integer is divisible by 3 if and only if the sum of its digits is divisible by 3. (b) Prove that it is divisible by 9 if and only if the sum of its digits is divisible by 9. (c) Prove that it is divisible by 1 1 if and only if the alternating sum of its...
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A Course in Number Theory and Cryptography

Neal Koblitz - Mathematics - 1994 - 235 pages
...decimal integer is divisible by 3 if and only if the sum of its digits is divisible by 3, and that it is divisible by 9 if and only if the sum of its digits is divisible by 9. 5. Prove that n 5 — n is always divisible by 30. 6. Suppose that in tiling a floor...
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A Course in Number Theory and Cryptography

Neal Koblitz - Mathematics - 1994 - 235 pages
...decimal integer is divisible by 3 if and only if the sum of its digits is divisible by 3, and that it is divisible by 9 if and only if the sum of its digits is divisible by 9. 5. Prove that n 5 — n is always divisible by 30. 6. Suppose that in tiling a floor...
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Five Hundred Mathematical Challenges

Edward J. Barbeau, Edward Barbeau, Murray S. Klamkin, W. O. J. Moser - Mathematics - 1995 - 227 pages
...under division by 9 is equal to that of a,. + a,._ 1 + . . + a 1 + a 0 . In particular, a number is divisible by 9 if and only if the sum of its digits is divisible by 9. SS •.. S SSS •.‘ S 15 . S .1. • S • • • S • • •‘‘ . SSSSS...
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A Mathematical Mosaic: Patterns & Problem Solving

Ravi Vakil - Mathematics - 1996 - 254 pages
...Thus M is divisible by 2 if and only if its last digit is divisible by 2! Divisibility by 9 i A number is divisible by 9 if and only if the sum of its digits is divisible by 9. (This method can easily be repeated. For example, to test if 1234567890 is divisible...
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Rings, Fields, and Vector Spaces: An Introduction to Abstract Algebra via ...

Bharath Sethuraman - Mathematics - 1996 - 192 pages
...99, and 999 as far as divisibility by 3 is concerned? Then, what does Lemma 1 .2 suggest?) (e) Prove that an integer is divisible by 9 if and only if the sum of its digits is divisible by 9. (0 Prove that an integer is divisible by 11 if and only if the difference between...
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Techniques of Problem Solving

Steven George Krantz - Mathematics - 1997 - 465 pages
...explain the reason in the following way: Notice that 45 is divisible by 9. We know that any number is divisible by 9 if and only if the sum of its digits is divisible by 9. In fact, any positive integer, divided by 9, has the same remainder as the remainder...
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An Introduction to Mathematical Reasoning

Peter J. Eccles - Mathematics - 1997 - 350 pages
...positive integer is written in decimal notation as n = Ofcofc-i... O2oiOO where 0 ^ <ц < 9. Prove that n is divisible by 9 if and only if the sum of its digits Ofc + uk-i + ... + G1 + 0 is divisible by 9. 4. Suppose that a positive integer is written in decimal...
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Mathemagic in the Classroom

Wade Sherard - Education - 1998 - 128 pages
...digits is 34, which yields 7 after casting out 9's And, 9 7=2, the missing digit. Property 4: A number is divisible by 9 if and only if the sum of its digits is divisible by 9 Let N be the serial number and let S be the “scrambled” number Suppose that N...
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