Asha visits the old-age home every 6 days and Bina every 8 days. Both visited on the circled date. On which date will they next visit on the same day AND have it be a Monday again?

They meet every LCM(6, 8) = 24 days. 24 = 3 weeks + 3 days, so each meeting moves the weekday on by 3. To come back to Monday we need 3$k$ to be a multiple of 7, i.e. $k=7$ meetings: $7\times 24=168$ days. 7 January + 168 days = 24 June (Jan 24 left, Feb 28, Mar 31, Apr 30, May 31 = 144; 168 − 144 = 24). (31 January is their next meeting, a Thursday.)