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The sum of the ones digit, double the tens digit, and four times the hundreds digit is divisible by 8. 34,152: 4 × 1 + 5 × 2 + 2 = 16. 9: The sum of the digits must be divisible by 9. [2] [4] [5] 2,880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9. Subtracting 8 times the last digit from the rest gives a multiple of 9. (Works because 81 is divisible by 9)
The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7. In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a ...
The difference of two squares can also be illustrated geometrically as the difference of two square areas in a plane. In the diagram, the shaded part represents the difference between the areas of the two squares, i.e. a 2 − b 2 {\displaystyle a^{2}-b^{2}} .
The corresponding logical symbols are "", "", [6] and , [10] and sometimes "iff".These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas ...
The actual difference is not usually a good way to compare the numbers, in particular because it depends on the unit of measurement. For instance, 1 m is the same as 100 cm, but the absolute difference between 2 and 1 m is 1 while the absolute difference between 200 and 100 cm is 100, giving the impression of a larger difference. [4]
The difference between explanations and arguments reflects a difference in the kind of question that arises. In the case of arguments, we start from a doubted fact, which we try to support by arguments. In the case of explanations, we start with an accepted fact, the question being why is this fact or what caused it.
Furthermore, if b 1, b 2 are both coprime with a, then so is their product b 1 b 2 (i.e., modulo a it is a product of invertible elements, and therefore invertible); [6] this also follows from the first point by Euclid's lemma, which states that if a prime number p divides a product bc, then p divides at least one of the factors b, c.
if p ≡ 3 (mod 8), then p is not a congruent number, but 2 p is a congruent number. if p ≡ 5 (mod 8), then p is a congruent number. if p ≡ 7 (mod 8), then p and 2 p are congruent numbers. It is also known that in each of the congruence classes 5, 6, 7 (mod 8), for any given k there are infinitely many square-free congruent numbers with k ...