⊗ means multiplication

6 x 7 = 42

42 ÷ 8 = 5 remainder 2 (42 = 5 x 8 + 2)

The remainder is the modulo, hence 42 modulo 8 is 2.

∴ y = 2

Modulo is a math operation that finds the remainder when one integer is divided by another.

The least number which a modulo is obtained is the remainder + the mod.

The least number in which the 3 (remainder) is obtained is 3 + 8 = 11

⇒ 7 + x = 11

x = 11 - 7

x = 4

Modulo is a math operation that finds the remainder when one integer is divided by another.

x = 7 because 17 = 2 x 7 + 3

The least is 9 dividing the number 1 time with a remainder of 7

4x = 9 x 1 + 7

4x = 9 + 7

4x = 16

Divide both sides by 4

x = $\frac{16}{4}$ = 4

4 (mod 8) = 8 x 1 + 4 = 12

Since y is greater than or equal to 10 (10 ≤ y, thus 10 is less than or equal to y), the minimum value should be 17 (7 + 10)

4 (mod 8) = 8 x 2 + 4 = 20

20 is more than the minimum value (17)

7 + y = 20

y = 20 - 7

y = 13

2x + 1 = 4(mod 7)

The least value which produces 4 in mod 7 is 1 x 7 + 4 = 11

2x + 1 = 11

2x = 11 - 1

2x = 10

Divide both sides by 2.

x = $\frac{10}{2}$ = 5

Note: modulo is the remainder after division

3 (mod 11) = 1 x 11 + 3 = 14

3 (mod 11) = 2 x 11 + 3 = 25

3 (mod 11) = 3 x 11 + 3 = 36

Using 14 = 3 (mod 11)

4x + 1 = 14

4x = 14 - 1

4x = 13

4 cannot divide 13 without a remainder. Hence we cannot use 14 = 3 (mod 11)

Using 25 = 3 (mod 11)

4x + 1 = 25

4x = 25 - 1

4x = 24

Divide both sides by 4

x = $\frac{24}{4}$

x = 6

⊗ means multiplication

6 x 7 = 42

42 ÷ 8 = 5 remainder 2 (42 = 5 x 8 + 2)

The remainder is the modulo, hence 42 modulo 8 is 2.

∴ y = 2

Modulo is a math operation that finds the remainder when one integer is divided by another.

The least number which a modulo is obtained is the remainder + the mod.

The least number in which the 3 (remainder) is obtained is 3 + 8 = 11

⇒ 7 + x = 11

x = 11 - 7

x = 4

Modulo is a math operation that finds the remainder when one integer is divided by another.

x = 7 because 17 = 2 x 7 + 3

The least is 9 dividing the number 1 time with a remainder of 7

4x = 9 x 1 + 7

4x = 9 + 7

4x = 16

Divide both sides by 4

x = $\frac{16}{4}$ = 4

4 (mod 8) = 8 x 1 + 4 = 12

Since y is greater than or equal to 10 (10 ≤ y, thus 10 is less than or equal to y), the minimum value should be 17 (7 + 10)

4 (mod 8) = 8 x 2 + 4 = 20

20 is more than the minimum value (17)

7 + y = 20

y = 20 - 7

y = 13

2x + 1 = 4(mod 7)

The least value which produces 4 in mod 7 is 1 x 7 + 4 = 11

2x + 1 = 11

2x = 11 - 1

2x = 10

Divide both sides by 2.

x = $\frac{10}{2}$ = 5