remainders

what numbers have a remainder of 1 when they are divided by 3?

what numbers have a remainder of 1 when they are divided by 2 or by 5?

what is the smallest number so that:

when it is put into bunches of 3
there is 1 left over

when it is put into bunches of 5
there is 2 left over

when it is put into bunches of 7
there is 3 left over?

there is a similar problem that was brought to my attention by David Wells, seemingly offered by Sun Tsu-Ching who worked on the Chinese remainder theorem (around the 4th century CE - I like the idea of students working on similar problems to their ancient ancestors):

• when you divide a number by 3, the remainder is 2
• when you divide it by 5, the remainder is 3
• when you divide it by 7, the remainder is 2

can students find the smallest and then the next biggest numbers?