Modular GCD LCM Quiz: Real-Life Math Applications

First find the sum: 87 plus 53 equals 140. Then find the remainder when 140 is divided by 11. 11 multiplied by 12 equals 132; 140 minus 132 equals 8. The remainder is 8. An alternative approach: 87 mod 11 equals 10 (since 11 multiplied by 7 equals 77 and 87 minus 77 equals 10), and 53 mod 11 equals 9 (since 11 multiplied by 4 equals 44 and 53 minus 44 equals 9). Adding the remainders: 10 plus 9 equals 19; 19 mod 11 equals 8. Both methods confirm the answer is 8.

First find the difference: 144 minus 37 equals 107. Then find the remainder when 107 is divided by 7. 7 multiplied by 15 equals 105; 107 minus 105 equals 2. The remainder is 2. An alternative approach: 144 mod 7 equals 4 (since 7 multiplied by 20 equals 140 and 144 minus 140 equals 4), and 37 mod 7 equals 2 (since 7 multiplied by 5 equals 35 and 37 minus 35 equals 2). Subtracting: 4 minus 2 equals 2. Both methods confirm the answer is 2.

The answer is True. The subtraction property of modular arithmetic states that remainders can be subtracted before or after reducing modulo n, as long as negative intermediate results are adjusted by adding n. For example, (14 minus 9) mod 5 equals 5 mod 5 equals 0, and (14 mod 5 minus 9 mod 5) mod 5 equals (4 minus 4) mod 5 equals 0. If the intermediate subtraction gives a negative result, adding n gives the correct non-negative remainder.

Using the multiplication property: (23 multiplied by 14) mod 5 equals (23 mod 5 multiplied by 14 mod 5) mod 5. 23 mod 5 equals 3 (since 5 multiplied by 4 equals 20 and 23 minus 20 equals 3). 14 mod 5 equals 4 (since 5 multiplied by 2 equals 10 and 14 minus 10 equals 4). So (3 multiplied by 4) mod 5 equals 12 mod 5 equals 2. Confirming directly: 23 multiplied by 14 equals 322; 322 divided by 5 gives remainder 2.

Reduce 15 modulo 7 first: 15 mod 7 equals 1 (since 7 multiplied by 2 equals 14 and 15 minus 14 equals 1). The equation becomes x plus 1 is congruent to 3 mod 7. Subtracting 1 from both sides: x is congruent to 2 mod 7. The least non-negative value of x is 2. Confirming: 2 plus 15 equals 17; 17 divided by 7 gives quotient 2 remainder 3.

First find the target value: 17 plus 8 equals 25; 25 mod 5 equals 0. Option A: (2 + 3) mod 5 equals 5 mod 5 equals 0 — matches. Option B: 17 mod 5 equals 2 and 8 mod 5 equals 3; (2 + 3) mod 5 equals 0 — matches. Option C: 26 mod 5 equals 1 (since 26 minus 25 equals 1) — does not match. Option D: (7 + 3) mod 5 equals 10 mod 5 equals 0 — matches. Options A, B, and D all equal 0, the same result as the original expression.

To find when both cyclists next align, compute LCM(8, 10). Using prime factorization: 8 equals 2 cubed and 10 equals 2 multiplied by 5. The LCM takes the highest power of each prime: 2 cubed multiplied by 5 equals 40. Both cyclists will next complete a lap together after 40 minutes. Confirming: 40 divided by 8 equals 5 laps for the first cyclist, and 40 divided by 10 equals 4 laps for the second.

The answer is False. If a is congruent to b mod n, then the multiplication property of modular arithmetic guarantees that a multiplied by a is congruent to b multiplied by b mod n. This is because the property states that if a is congruent to b and c is congruent to d mod n, then a multiplied by c is congruent to b multiplied by d mod n. Applying this with c equal to a and d equal to b gives a squared is congruent to b squared mod n.

To find when both tanks are next refilled together, compute LCM(12, 16). Using prime factorization: 12 equals 2 squared multiplied by 3 and 16 equals 2 to the fourth. The LCM takes the highest power of each prime: 2 to the fourth multiplied by 3 equals 48. Both tanks will next be refilled together after 48 minutes. Confirming: 48 divided by 12 equals 4 refill cycles, and 48 divided by 16 equals 3 refill cycles.

The greatest group size that divides both 90 and 75 exactly is GCD(90, 75). Using the Euclidean Algorithm: 90 divided by 75 gives remainder 15; 75 divided by 15 gives remainder 0. GCD equals 15. Confirming: 90 divided by 15 equals 6 groups of boys, and 75 divided by 15 equals 5 groups of girls, with no students left over.

The greatest piece length is GCD(65, 91). Using the Euclidean Algorithm: 91 divided by 65 gives remainder 26; 65 divided by 26 gives remainder 13; 26 divided by 13 gives remainder 0. GCD equals 13 metres. Confirming: 65 divided by 13 equals 5 pieces, and 91 divided by 13 equals 7 pieces, with no rope left over.

The greatest number of bags is GCD(72, 48). Using the Euclidean Algorithm: 72 divided by 48 gives remainder 24; 48 divided by 24 gives remainder 0. GCD equals 24. Each red pen bag contains 72 divided by 24 equals 3 pens, and each blue pen bag contains 48 divided by 24 equals 2 pens. Confirming: 24 red bags plus 24 blue bags gives 48 bags in total with no pens left over.

The answer is True. This is a standard property of the GCD. Both GCD(a, b) and GCD(a, a minus b) give the same result because any common divisor of a and b also divides a minus b, and any common divisor of a and a minus b also divides b. For example, GCD(12, 8) equals 4, and GCD(12, 4) also equals 4. This property is what makes the Euclidean Algorithm work.

First find the sum: 99 plus 46 equals 145. Then find the remainder when 145 is divided by 13. 13 multiplied by 11 equals 143; 145 minus 143 equals 2. The remainder is 2. Alternatively: 99 mod 13 equals 8 (since 13 multiplied by 7 equals 91 and 99 minus 91 equals 8), and 46 mod 13 equals 7 (since 13 multiplied by 3 equals 39 and 46 minus 39 equals 7). Adding: 8 plus 7 equals 15; 15 mod 13 equals 2. Both methods confirm the answer.

First find the target value: 22 plus 18 equals 40; 40 mod 8 equals 0. Option A: 40 mod 8 equals 0 — matches. Option B: 22 mod 8 equals 6 and 18 mod 8 equals 2; (6 + 2) mod 8 equals 8 mod 8 equals 0 — matches. Option C: (5 + 4) mod 8 equals 9 mod 8 equals 1 — does not match. Option D: 48 mod 8 equals 0 (since 48 divided by 8 equals 6 exactly) — matches. Options A, B, and D all equal 0.

To find when both candles finish together, compute LCM(4, 6). Using prime factorization: 4 equals 2 squared and 6 equals 2 multiplied by 3. The LCM takes the highest power of each prime: 2 squared multiplied by 3 equals 12. Both candles will next finish a complete burn at the same time after 12 hours. Confirming: 12 divided by 4 equals 3 complete burns for the first candle, and 12 divided by 6 equals 2 complete burns for the second.

Using prime factorization: 10 equals 2 multiplied by 5 and 14 equals 2 multiplied by 7. The LCM takes the highest power of each prime factor: 2 multiplied by 5 multiplied by 7 equals 70. So LCM(10, 14) equals 70. Confirming: 70 divided by 10 equals 7 (so 10 divides 70 exactly), and 70 divided by 14 equals 5 (so 14 divides 70 exactly). No smaller positive integer is divisible by both 10 and 14.

The answer is False. GCD is symmetric for all pairs of positive integers, not just when a equals b. GCD(a, b) always equals GCD(b, a) because the set of common divisors of a and b is the same regardless of order. For example, GCD(12, 8) equals 4 and GCD(8, 12) also equals 4. The GCD depends only on which numbers are involved, not the order in which they are written.

The largest square tile has a side length equal to GCD(110, 66). Using the Euclidean Algorithm: 110 divided by 66 gives remainder 44; 66 divided by 44 gives remainder 22; 44 divided by 22 gives remainder 0. GCD equals 22 cm. Confirming: 110 divided by 22 equals 5 tiles across, and 66 divided by 22 equals 3 tiles down, covering the wall with 15 tiles total and no cutting needed.

Using the identity a multiplied by b equals GCD(a, b) multiplied by LCM(a, b): 360 equals 6 multiplied by LCM(a, b). Dividing both sides by 6 gives LCM(a, b) equals 360 divided by 6 equals 60. Confirming: if a equals 12 and b equals 30, then GCD(12, 30) equals 6 and LCM(12, 30) equals 60, and 12 multiplied by 30 equals 360, consistent with all given values.