. Prove the following: Let a, b, c, n E Z, where n 2 2. If ac = bc (mod n) and gcd(c, n) = 1,then a = b (mod n)

To prove that a = b (mod n), we need to show that n divides (a - b).

Since ac = bc (mod n), we have n divides (ac - bc) = c(a - b).

Since gcd(c, n) = 1, we know that n and c are coprime, which means that n does not divide c.

Therefore, n must divide (a - b), which means that a = b (mod n).

Hence, the statement is proved.

相关问题

12 Prove and disprove (a) Prove or disprove the following claim: 16n = (b) Prove or disprove the following claim: = O(16n)

(a) This claim is incomplete as there is no value or expression given after the equals sign. Therefore, it cannot be proven or disproven without additional information.

(b) This claim is true. To prove it, we need to show that there exists a constant c and an n0 such that 16n ≤ c16n for all n ≥ n0. Let c = 1 and n0 = 1. Then, 16n ≤ 16n for all n ≥ 1, which satisfies the definition of O(16n). Therefore, the claim is proven.

def get_queryset(self, request): qs = warehousePeople.objects.all() if 'name' in request.GET and request.GET['name'] != 'none': # 条件查询 qs = qs.filter(name=request.GET['name']) qs1 = qs.none() if 'subjectMatter' in self.list_display: self.list_display.remove('subjectMatter') if 'apply_prove' not in self.list_display: self.list_display.append('apply_prove') for obj in qs: if qs1.filter(month=obj.month, warehouse=obj.warehouse.id).exists(): break else: infoLi = qs.filter(month=obj.month, warehouse=obj.warehouse.id) first_obj = infoLi.first() infoLiOne = qs.filter(pk=first_obj.pk) qs1 |= infoLiOne else: if 'subjectMatter' not in self.list_display: self.list_display.append('subjectMatter') if "apply_prove" in self.list_display: self.list_display.remove('apply_prove') if 'name' in request.GET and request.GET['name'] != 'none': return qs1 else: return qs 代码中如何使用annotate方法动态的为qs1添加动态字段apply_prove？

可以使用annotate方法来为qs1添加动态字段apply_prove，具体代码如下：

from django.db.models import Case, When, IntegerField

qs1 = qs.none().annotate(
    apply_prove=Case(
        When(warehouse__isnull=True, then=0),
        When(month__isnull=True, then=0),
        default=1,
        output_field=IntegerField(),
    )
)

这段代码中，我们使用了annotate方法来为qs1添加一个名为apply_prove的动态字段。annotate方法可以接收多个参数，每个参数都是一个表达式，用来为查询集中每个对象添加一个字段。在这个例子中，我们使用了Case表达式来根据warehouse和month是否为空来计算apply_prove的值。具体来说，当warehouse或month为空时，apply_prove的值就为0，否则就为1。最后，我们指定了这个动态字段的输出类型为IntegerField。