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When the Mods opened, they were intended as temporary housing. After his speech, Burns made only a couple of stops on campus.
What is a mod mod#
Just how beloved are the Mods? In his 2002 Commencement address, the former NATO ambassador Nicholas Burns ’78 mentioned exactly one BC building: the Mod where he had lived (36A). “I always envisioned that that’s where I’d live,” he said. And the way he saw it, Mod living was practically a birthright. For Bryan, applying to BC was preordained, a no-brainer. He was a legacy who’d basically grown up on campus, attending football, basketball, and hockey games. I recall a post-lotto conversation with one of my disappointed roommates. When fate inevitably denied a space to the majority of us, there was heartache. As the drawing rolled around during my senior year, I remember classmates saying little prayers, hoping divine intervention would deliver them into the Mods.
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You could score a Mod only by selecting five potential roommates and then hoping your number was drawn in the annual housing lottery. So, despite the fact that they were small and dingy, there wasn’t a single person I knew who didn’t want to call one of the ruddy-brown huts home. As one friend used to say-without a whiff of hyperbole-“Everything happens in the Mods.” Sure, you get in through sheer luck, but the perception has always been that people who live in the Mods are part of a special club. Being a Mod resident carries with it a certain cachet. If n is composite, there exists a subgroup of the multiplicative group, called the "group of false witnesses", in which the elements, when raised to the power n − 1, are congruent to 1 modulo n.The prefab, two-story Mods (short for modular housing) were installed in 1970, and they’ve developed a certain mystique, both because they are somewhat exclusive-there’s room for only 438 seniors to live there, about a fifth of the class-and an epicenter of campus culture. It is denoted Z / n Z and is equal to it if and only if the group is cyclic.
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The set of (congruence classes of) integers modulo n with the operations of addition and multiplication is a ring. Notice that the equation ax + ny = 1 implies that x is coprime to n, so the multiplicative inverse belongs to the group. It exists precisely when a is coprime to n, because in that case gcd( a, n) = 1 and by Bézout's lemma there are integers x and y satisfying ax + ny = 1. This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity.įinally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ 1 (mod n).
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Integer multiplication respects the congruence classes, that is, a ≡ a' and b ≡ b' (mod n) implies ab ≡ a'b' (mod n). Since gcd( a, n) = 1 and gcd( b, n) = 1 implies gcd( ab, n) = 1, the set of classes coprime to n is closed under multiplication. Thus the notion of congruence classes modulo n that are coprime to n is well-defined. Integers in the same congruence class a ≡ b (mod n) satisfy gcd( a, n) = gcd( b, n), hence one is coprime to n if and only if the other is. Indeed, a is coprime to n if and only if gcd( a, n) = 1. It is a straightforward exercise to show that, under multiplication, the set of congruence classes modulo n that are coprime to n satisfy the axioms for an abelian group.