I mean to ask that after sex is wudhu enough or is gusul necessary for offering of prayers?
My husband often says he can't offer prayer as he has not taken bath!
If a man and woman (hopefully spouses) had intercourse they are asked to perform ghusl (once the glans has fully penetrated the vagina) to be able to pray -as this is what is called the state of janabah-. In case that they can't do it because of a valid reason such as the lack of water they are allowed to perform tayamum. This is simply what the Quran says:
O you who have believed, do not approach prayer while you are intoxicated until you know what you are saying or in a state of janabah, except those passing through [a place of prayer], until you have washed [your whole body]. And if you are ill or on a journey or one of you comes from the place of relieving himself or you have contacted women and find no water, then seek clean earth and wipe over your faces and your hands [with it]. Indeed, Allah is ever Pardoning and Forgiving. (4:43)
O you who have believed, when you rise to [perform] prayer, wash your faces and your forearms to the elbows and wipe over your heads and wash your feet to the ankles. And if you are in a state of janabah, then purify yourselves. But if you are ill or on a journey or one of you comes from the place of relieving himself or you have contacted women and do not find water, then seek clean earth and wipe over your faces and hands with it. Allah does not intend to make difficulty for you, but He intends to purify you and complete His favor upon you that you may be grateful. (5:6)
Istinja' is only related to wudu' and is only necessary If you had urinated or defecated (and you are clean of janabah, menses etc.)!
Is istinja part of wudu?
Is istinja' necessary for ablution?
What kind of istinja is permissible in Islam?
Is wudu valid after intercourse?
Can we pray after sex if we didn't ejaculate?
Is it permissable to pray after intercourse?
How to attain full purity? Please see details
Is it mandatory to wash hair after having sex?
So basically your inquiry has answers on the site.
Both of you need to perform ghusl and shouldn't waste time and be lazy to pray!