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Does one need to search/clean for even a crumb of Chametz?
One must search for even a crumb of Chametz, and destroy it. This applies even if one has already nullified his Chametz, or plans to nullify it before Pesach. If, however, the Chametz is dirty and one has already nullified his Chametz, or plans to do so before the 6th hour, then if the Chametz is less than the size of a Kezayis, one is not required to search for this Chametz or destroy it. Likewise, if the less than Kezayis nullified piece is stuck onto a wall or vessel, then it does not need to be destroyed, so long as within the entire room or vessel there isn’t a Kezayis amount of scattered Chametz, will be explained in Halacha 4.
Crumbs of Chametz found in areas where people walk: On the floor, in areas that people walk, one only has to search/clean slightly large crumbs of Chametz. However, thin crumbs of Chametz, does not need to be cleaned or removed from such areas. However, if one sees them on the floor after the 6th hour on Erev Pesach, then one must gather them and destroy them, [if the crumbs are more than a Kezayis, or Bittul was not performed].
Based on the above law, it is not necessary to clean or purchase a new broom stick for Pesach, as even if it contains Chametz, it is less than a Kezayis, is dirty, and are nullified. Nonetheless, some are particular to do so.
Based on the above law, it is not necessary to change the vacuum bag before Pesach, as even if it contains Chametz, the individual pieces are less than a Kezayis, are dirty, and are nullified. Nonetheless, the custom is to do so.
 Admur 433:13; 442:28; 444:9; 446:3; Chayeh Adam 119:10; Poskim in M”B 442:33
Background: In the end of 433:13 Admur says that one needs to check for even “Shum Pirur Chametz” and in 442:28 that Rabbinically one may not own even less than a Kezayis. In 444:9 Admur rules that one needs to destroy the small leftover crumbs when Erev Pesach falls on Shabbos, and in 446:3 that on Pesach one must destroy even less than a Kezayis of Chametz which one has done bittul to.
Other opinions: Some Poskim rule one is not required to each for a crumb of Chametz which is less than a Kezayis. [See Poskim in Nitei Gavriel 3 Teshuvas 1]
 The reason: As the Sages decreed against owning even less than a Kezayis of Chametz, lest one come to own a Kezayis. [Admur 442:28] This decree only applies in a case that one did not nullify his Chametz before Pesach, or does not plan to do so. In such a case, one transgresses a Rabbinical prohibition of Baal Yiraeh over such Chametz, even though it is less than a Kezayis. If, however, one nullified the Chametz, then it follows the law and reason explained next. [Admur Kuntrus Acharon 446:1]
 Admur 442:28; 446:3
The reason: One must search for even less than a Kezayis crumb of Chametz which is anyways nullified because one may come to forget and inadvertently eat it on Pesach. [Admur 442:28] It, however, is not prohibited due to a Rabbinical Baal Yiraeh prohibition [as if it was already nullified it would be a double Dirabanan, which is a decree upon a decree]. [Kuntrus Acharon 446:1] Due to this, one does not say a blessing of Bedikas Chametz if one only has suspicion that less than a Kezayis of Chametz has remained to be checked for. [Admur 432:5] Likewise, when burning less than a Kezayis of nullified Chametz on Pesach, a blessing may never be said. [Admur 446:3; Kuntrus Acharon 446:1]
Other opinions: Some Poskim rule it is permitted even Rabbinically to own less than a Kezayis of Chametz which has been nullified before Pesach. [Opinion brought in Admur 446:3; M”A 442:12 in his understanding of: Abayei in Pesachim 45a; Riy in Tosafos ibid; Rosh 3:2; Tur 442 in name of Rosh; Peri Chadash 442:7]
 Admur 442:28; M”A 442:10; Yireim 301; Hagahos Maimanis 2:15 Taf; Vetzaruch Iyun if one finds a less than a Kezayis piece of Chametz which was nullified, if it sufficed for one to dirty it to the point that people will not come to eat it, or to simply throw it out of one’s house on Pesach.
The reason: As less than a Kezayis of nullified Chametz is not prohibited due to a Rabbinical owning prohibition of Baal Yiraeh, but rather simply due to worry that one may come to eat it. Now, since when the Chametz is dirty there is no need to suspect that one will come to eat it, therefore, there is no need to get rid of it. [See Admur Kuntrus Acharon 446:1]
Must one actively nullify a less than Kezayis piece? All insignificant size crumbs of baked Chametz which are less than a Kezayis is considered automatically nullified, and thus it is not necessary to actively nullify it before Pesach. However, a significant size crumb of baked Chametz even if less than a Kezayis is not considered automatically nullified, and thus must actively be nullified before Pesach. [See Admur 434:6; 460:5] A piece of dough is considered significant and must be actively nullified before Pesach. [Admur ibid]
 Admur 442:28
 Admur 444:9
 The reason: Even if one sees the crumbs lying on the ground, one is not required to gather them and burn them if they are in an area where people step on, being that they end up getting destroyed on their own through stepping on them. [Admur ibid]
 See Nitei Gavriel 76:6