What is the factor by which the reward for righteous deeds in Ramadan is multiplied?

Because Ramadan is of such a high status, acts of obedience during the month are of immense virtue and earn a greatly multiplied reward, and the sin of acts of disobedience during this month is greater and more serious than sins at other times. So the Muslim should make the most of this blessed month by doing acts of worship and righteous deeds, and giving up bad deeds, in the hope that Allah, may He be glorified and exalted, will bless him by accepting those deeds and will enable him to adhere to the truth. But the bad deed always incurs an equal burden of sin, and is never multiplied, either in Ramadan or at other times, whereas a good deed receives a multiple reward, tenfold up to many times more than that.

End quote from Majmoo‘ Fataawa Ibn Baaz (15/447) 

The multiplication of reward for good deeds – in Ramadan and at other times – is multiplication in terms of quantity and quality. What is meant by quantity is the number, so a good deed earns a more than tenfold reward. What is meant by quality is that the reward is made greater and more abundant. As for the bad deed, it is multiplied in terms of quality only, in other words the level of sin and the punishment is greater. 

Please see the answer to question no. 38213

Secondly: 

Islamic teachings explain these multiplications with regard to some deeds and some times. 

With regard to Laylat al-Qadr, Allah, may He be glorified and exalted, says (interpretation of the meaning): “The night of Al-Qadr (Decree) is better than a thousand months” [al-Qadr 97:3]. That is, worship on that night is better than worshipping for a thousand months. 

Shaykh as-Sa‘di (may Allah have mercy on him) said: 

That is, it is equal in virtue to one thousand months, so a deed that is done on that night is better than a deed that is done for one thousand months without Laylat al-Qadr. This is something that is utterly astounding and dazzles the mind, as Allah, may He be Blessed and exalted, has bestowed upon this ummah, which is weak in terms of strength and resolve, a night on which a good deed is equal to and exceeds one thousand months, when the lifespan of a long-lived man is only eighty odd years.

End quote from Tafseer as-Sa‘di (p. 931) 

With regard to performing ‘Umrah in Ramadan, it was narrated that the Prophet (blessings and peace of Allah be upon him) said: “ ‘Umrah in Ramadaan is equivalent (in reward) to Hajj – or to Hajj with me.”

Narrated by al-Bukhaari (1863) and Muslim (1256). 

Al-Qaari (may Allah have mercy on him) said: 

“equivalent (in reward) to Hajj” means: it is equal in terms of reward.

End quote from Mirqaat al-Mafaatih (5/1742) 

Undoubtedly the reward for Hajj is many times greater than the reward for ‘umrah, and Hajj is one of the pillars of Islam. There are some hadiths that speak of this superiority, but they are da‘eef (weak) hadiths and cannot be taken as evidence. One of them is the hadith that says: “O people, there has come to you a great month, a month in which there is a night that is better than a thousand months. Allah has made fasting this month obligatory, and has made prayer during its nights voluntary. Whoever draws close to Him by doing a good deed, it will be as if he had done an obligatory deed at other times, and whoever performs an obligatory deed during it will be like one who performed seventy obligatory deeds at other times.” 

Al-Albaani said in ad-Da‘eefah (871): It is munkar (odd). See the answer to question no. 21364.

There are reports from some of the salaf (early generations) which speak of the factor by which the reward is multiplied, and they are reports that may be quoted but cannot be taken as evidence. 

Abu Bakr ibn Abi Maryam narrated from his shaykhs that they used to say: When the month of Ramadan comes, then spend generously during it, because spending during it brings a multiple reward, like spending in Allah’s cause, and saying tasbeeh in it is better than saying a thousand tasbeehs at other times. 

An-Nakha‘i (may Allah have mercy on him) said:

Fasting one day in Ramadan is better than fasting a thousand days, and saying tasbeeh in it is better than saying a thousand tasbeehs, and one rak‘ah in it is better than a thousand rak‘ahs.

End quote from Lataa’if al-Ma‘aarif by Ibn Rajab (p. 151) 

Ibn Abi ad-Dunya narrated in Fadaa’il Ramadan (p. 51) that az-Zuhri said: One tasbeeh in Ramadan is better than saying a thousand tasbeehs at other times. End quote. 

Conclusion: 

With regard to the factor by which the reward for good deeds in Ramadan is multiplied, Islamic teachings do not say anything in detail, but the Muslim should strive hard to do righteous deeds in Ramadan so that he may attain those virtues. 

And Allah knows best.