418531 ภาคต้น 2552/โจทย์ปัญหาการวิเคราะห์เชิงการจัด/เฉลยข้อ 1

ในปัญหาข้อนี้ให้

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 969640435
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:23 GMT


Sensitive client information

IP address: 158.108.32.49

$A_{n,k}\,$ แทนเซตของจำนวนเต้มจาก 1 ถึง $n\,$ ที่

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 953497116
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:23 GMT


Sensitive client information

IP address: 158.108.32.49

$k\,$ หารลงตัว เราได้ว่า

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 964238082
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:24 GMT


Sensitive client information

IP address: 158.108.32.49

$A_{n,k}={\bigg \lfloor }{\frac {n}{k}}{\bigg \rfloor }$

ให้

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 964270780
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:24 GMT


Sensitive client information

IP address: 158.108.32.49

$B_{k}\,$ แทนเซตของจำนวนเต็มจาก 1000 ถึง 9999 ที่ k หารลงตัว เราได้ว่า

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 936921332
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:24 GMT


Sensitive client information

IP address: 158.108.32.49

$B_{k}=A_{9999,k}-A_{999,k}\,$ ฉะนั้น

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 938231161
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:25 GMT


Sensitive client information

IP address: 158.108.32.49

$|B_{k}|=|A_{9999,k}|-|A_{999,k}|={\bigg \lfloor }{\frac {9999}{k}}{\bigg \rfloor }-{\bigg \lfloor }{\frac {999}{k}}{\bigg \rfloor }\,$

ข้อ 1

เซตของจำนวนเต็มจาก 1000 ถึง 9999 ที่ 9 หารลงตัวคือ

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 946617546
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:25 GMT


Sensitive client information

IP address: 158.108.32.49

$B_{9}\,$ ซึ่งมีสมาชืก

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 968658398
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:25 GMT


Sensitive client information

IP address: 158.108.32.49

${\bigg \lfloor }{\frac {9999}{9}}{\bigg \rfloor }-{\bigg \lfloor }{\frac {999}{9}}{\bigg \rfloor }=1111-111=1000\,$ ตัว

ข้อ 2

เซตของจำนวนเต็มจาก 1000 ถึง 9999 ที่เป็นจำนวนคู่คือ

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 972457441
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:25 GMT


Sensitive client information

IP address: 158.108.32.49

$B_{2}\,$ ซึ่งมีสมาชืก

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 910360174
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:26 GMT


Sensitive client information

IP address: 158.108.32.49

${\bigg \lfloor }{\frac {9999}{2}}{\bigg \rfloor }-{\bigg \lfloor }{\frac {999}{2}}{\bigg \rfloor }=4999-499=4500\,$ ตัว

ข้อ 3

เซตของจำนวนเต็มจาก 1000 ถึง 9999 ที่ 3 หารลงตัว คือ

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 965875454
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:26 GMT


Sensitive client information

IP address: 158.108.32.49

$B_{3}\,$ ซึ่งมีสมาชิก

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 910261385
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:26 GMT


Sensitive client information

IP address: 158.108.32.49

${\bigg \lfloor }{\frac {9999}{3}}{\bigg \rfloor }-{\bigg \lfloor }{\frac {999}{3}}{\bigg \rfloor }=3333-333=3000\,$ ตัว

ดังนั้นเซตของจำนวนเต็มจาก 1000 ถึง 999 ที่ 3 หารไม่ลงตัวจึงมีสมาชิกทั้งหมด

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 971737410
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:27 GMT


Sensitive client information

IP address: 158.108.32.49

$(9999-1000+1)-3000=9000-3000=6000\,$ ตัว

ข้อ 4

เซตของจำนวนเต็มจาก 1000 ถึง 9999 ที่หารด้วย 5 หรือ 7 ลงตัวคือ

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 930109901
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:27 GMT


Sensitive client information

IP address: 158.108.32.49

$B_{5}\cup B_{7}\,$

เรารู้ว่า

Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 974323788 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:27 GMT Sensitive client information IP address: 158.108.32.49 $|B_{5}\cup B_{7}|\,$	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 967577866 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:28 GMT Sensitive client information IP address: 158.108.32.49 $|B_{5}|+|B_{7}|-|B_{5}\cap B_{7}|\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 955166802 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:28 GMT Sensitive client information IP address: 158.108.32.49 $|B_{5}|+|B_{7}|-|B_{35}|\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 965875498 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:29 GMT Sensitive client information IP address: 158.108.32.49 ${\bigg (}{\bigg \lfloor }{\frac {9999}{5}}{\bigg \rfloor }-{\bigg \lfloor }{\frac {999}{5}}{\bigg \rfloor }{\bigg )}+{\bigg (}{\bigg \lfloor }{\frac {9999}{7}}{\bigg \rfloor }-{\bigg \lfloor }{\frac {999}{7}}{\bigg \rfloor }{\bigg )}-{\bigg (}{\bigg \lfloor }{\frac {9999}{35}}{\bigg \rfloor }-{\bigg \lfloor }{\frac {999}{35}}{\bigg \rfloor }{\bigg )}\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 943637165 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:29 GMT Sensitive client information IP address: 158.108.32.49 $(1999-199)+(1428-142)-(285-28)\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 958475931 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:30 GMT Sensitive client information IP address: 158.108.32.49 $1800+1286-257\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 947470382 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:31 GMT Sensitive client information IP address: 158.108.32.49 $2829\,$ ตัว

ข้อ 5

เซตของจำนวนเต็มจาก 1000 ถึง 9999 ที่ 5 และ 7 หารลงตัวคือ

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 951891406
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:31 GMT


Sensitive client information

IP address: 158.108.32.49

$B_{5}\cap B_{7}=B_{35}\,$ ซึ่งมีสมาชิก ${\bigg \lfloor }{\frac {9999}{35}}{\bigg \rfloor }-{\bigg \lfloor }{\frac {999}{35}}{\bigg \rfloor }=285-28=257\,$ ตัว

ดังนั้นเซตของจำนวนเต็มจาก 1000 ถึง 999 ที่ 5 และ 7 หารไม่ลงตัวจึงมีสมาชิกทั้งหมด

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 945241042
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:32 GMT


Sensitive client information

IP address: 158.108.32.49

$(9999-1000+1)-257=9000-257=8743\,$ ตัว

ข้อ 6

เซตของจำนวนเต็มจาก 1000 ถึง 9999 ที่หารด้วย 5 หรือ 7 หรือ 11 ลงตัวคือ

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 946191624
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:32 GMT


Sensitive client information

IP address: 158.108.32.49

$B_{5}\cup B_{7}\cup B_{11}\,$

เรารู้ว่า

Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 934561637 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:32 GMT Sensitive client information IP address: 158.108.32.49 $|B_{5}\cup B_{7}\cup B_{11}|\,$	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 956609487 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:33 GMT Sensitive client information IP address: 158.108.32.49 $|B_{5}|+|B_{7}|+|B_{11}|-|B_{5}\cap B_{7}|-|B_{5}\cap B_{11}|-|B_{7}\cap B_{11}|+|B_{5}\cap B_{7}\cap B_{11}|\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 970360869 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:33 GMT Sensitive client information IP address: 158.108.32.49 $|B_{5}|+|B_{7}|+|B_{11}|-|B_{35}|-|B_{55}|-|B_{77}|+|B_{385}|\,$
	$=\,$
		Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 952317861 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 29 Oct 2026 14:51:34 GMT Sensitive client information IP address: 158.108.32.49 $(1999-199)+(1428-142)+(909-90)-(285-28)-(181-18)-(129-12)+(25-2)\,$
		ตัว

ข้อ 7

เซตของจำนวนเต็มจาก 1000 ถึง 9999 ที่ 5 และ 7 หารลงตัว คือ ซึ่งมีสมาชิก ตัว

เซตของจำนวนเต็มจาก 1000 ถึง 9999 ที่ 11 หารลงตัว คือ ซึ่งมีสมาชิก ตัว

และเซตของจำนวนเต็มจาก 1000 ถึง 999 ที่ 11 หารไม่ลงตัวจึงมีสมาชิกทั้งหมด ตัว

ดังนั้นเซตของจำนวนเต็มจาก 1000 ถึง 999 ที่ 5 และ 7 หารลงตัว แต่ 11 หารไม่ลงตัวจึงมีสมาชิกทั้งหมด ตัว