Find the greatest number which on dividing 60 and 90 leaves remainders 2 and 3 respectively.

When dividing 60, the remainder is 2.

When dividing 90, the remainder is 3.

If a number N leaves a remainder of 2 when dividing 60, then 60 − 2 = 58 is divisible by N.

Similarly, if N leaves a remainder of 3 when dividing 90, then 90 − 3 = 87 is divisible by N.

HCF of 58 and 87 = 29

Thus, The greatest number N is 29.