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#21 Digit gangs

Set by Katie Steckles

If we use the digits 0-9 in place of the letters A-J (in some order) in the numbers ABC, DE, FGH and IJ, we can create four numbers to use in calculations. For example, the numbers could be 134, 50, 289 and 67, or 013, 56, 289 and 74.

What is the largest total you can make for ABC + DE + FGH + IJ by assigning the digits?

Can you assign the digits to maximise the value of (ABC 脳 DE) + (FGH 脳 IJ)? What is the largest total you can get?

How can you assign the digits to make the value of (ABC 脳 DE) 鈥� (FGH 脳 IJ) as close to zero as possible?

Solution next week

#20 Factor-finding mission

Solution

The first four primes are 2, 3, 5 and 7, and their product is 210, giving it four prime factors. Every other factor of 210 can be made by multiplying some of the prime factors together. Each prime factor can be either included or not in the multiplication, so there are 2 脳 2 脳 2 脳 2 = 16 factors: 1 (the case when no prime factors are included), 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210). The next number with16 factors is surprisingly close 鈥� 216 is 2³ 脳 3³, which means its factors are made up of 0, 1, 2 or 3 of each of the two prime factors. This gives 4 脳 4 = 16 factors (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108 and 216). The lowest number with 16 factors is 120, 2³ 脳 3 脳 5.