Please refer to the following hadith narration:
https://sunnah.com/bukhari:969
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This hadith narration does indicate importance of first ten days of Zulhajja and this importance is primarily because of Hajj and the sunnah of animal sacrifice as well – this is also the reason that in the month of animal sacrifice (first ten day of zulhajja), those who intend to sacrifice an animal are advised to not cut their nails or hair before they have offered sacrifice (Please see: Discussion 71450). However, this should not mean that these ten days are superior to any other blessed days fixed by Allah, for example, the month of Ramdhan, the last ten nights of Ramdhan and the night of Laylat-ul Qadr etc. They all have their importance in the sight of Allah, in one way or another and there is a reason for fixing such blessed days which is to provide a window of ‘ruju‘ to people to have an opportunity to revert to Allah and seek his blessings and his forgiveness (Please see: Discussion 45822).