HW 2025-02-14 WS

[AOPS INT 12.25]

How many integers between 1000 and 3000 inclusive are congruent to 5 (mod 7)?

[AOPS INT 12.26]

For how many values of n, where 40 ≤ n ≤ 80 is n ≡ -n (mod 12)?

[AOPS INT 12.27]

Four jars of jelly beans are individually repackaged into packs of 12 jelly beans. After the maximum number of packs are filled from each of these jars, the numbers of remaining jelly beans in the jars are 3, 5, 7, and 11. If the four jars had been combined before repackaging and the maximum number of packs were filled, how many jelly beans would have been left out of the 12 packs?

[AOPS INT 12.28]

Two positive integers leave remainders of 6 and 13 when divided by 14.

Find the remainder when their sum is divided by 14.

[AOPS INT 12.28]

Two positive integers leave remainders of 6 and 13 when divided by 14.

Find the remainder when their difference is divided by 14.

[AOPS INT 12.28]

Two positive integers leave remainders of 6 and 13 when divided by 14.

Find the remainder when their product is divided by 14.

[AOPS INT 12.28]

Two positive integers leave remainders of 6 and 13 when divided by 14.

Find the remainder when their product is divided by 7.

[AOPS INT 12.30]

Find the remainder when 7²⁵⁵ is divided by 11.

[AOPS INT 12.31]

Find the remainder when 9⁴² - 5⁴² is divided by 7.

[AOPS INT 12.33]

Find the last two digits of 99²⁰⁰⁵.