Finding Common Ground: questions with answers

The HCF is 2. It divides all three numbers and is the greatest such divisor. The number 6 divides 12 and 18, but not 20. The number 1 is common but not greatest. The number 180 is a common multiple, not a common factor.

The HCF is 14, the greatest positive integer dividing both 28 and 42. The number 28 divides only the first input. The number 84 is a common multiple. Although 1 divides both, these inputs have a greater common factor.

The counts of 2 are 1 and 2. The HCF uses the smaller count, 1, because it must divide both numbers. Adding the counts or using the larger count includes copies that one number does not contain.

Use only primes present in both factorizations, with the smaller occurrence count for each. Their product is 6. Using larger counts gives 432, which does not divide 378. The number 210 includes a prime absent from one input, so it cannot be a common factor.

The HCF is m. Since 7m is a multiple of m, m divides both numbers. No factor of m is larger than m. So the HCF equals the smaller number.

The HCF is 2. Any common divisor of 38 and 40 divides their difference, 2. Both numbers are even, so 2 divides both and is their greatest common divisor. They are consecutive even numbers, not consecutive integers.

Every completed route gives 2 × 3 × 5 × 5, possibly in a different order. This product contains 4 prime factors, counting repetitions. A different starting split changes neither the primes nor their repetitions. The number 1 is not prime, so it is not included.

Leave out 1 because it is not prime. Multiplying by 1 leaves the product unchanged. Every remaining factor is prime, so the prime factorisation is 120 = 2 × 2 × 2 × 3 × 5.

The day number must be a multiple of both 12 and 18. Their LCM is 36, so both events happen together every 36 days. The first three later day numbers are 36, 36 × 2 = 72, and 36 × 3 = 108. This pattern continues while both events keep repeating. Their HCF, 6, is too small to be a multiple of either number.

Both 8 and 12 divide M, so both also divide 2 × M. Since M is positive, 2 × M is larger than M. This works for every positive common multiple, so there is no largest one.

There are 32 distinct positive factors. Form products using the available prime factors, including no factors for 1 and all factors for 840. Count each resulting number only once. The 6 prime occurrences and 4 distinct primes are different counts.

225 = (3 × 3) × (5 × 5). The product 3 × 3 is a subpart of the prime factorisation, so it is a factor of 225. Each other product uses an unavailable prime or repeats a prime too many times.

The counts of 3 are 2 and 1. The LCM needs the larger count, 2, so both numbers divide it. A smaller count is insufficient; adding the counts supplies unnecessary copies.

96 = 8 × 12 and 360 = 8 × 45. So 8 divides both numbers exactly. A common divisor used in the division method may be composite. Of the other choices, 16 divides only 96, and 45 divides only 360.

For two positive integers, HCF × LCM equals their product. Here, 90720 = 6 × 15120. Dividing the product by the LCM gives the whole number 6, with remainder 0. So the LCM is a factor of the product.