Daily Math Puzzle: 2026-06-19
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2026-06-19
Maya went shopping and bought several identical items, each costing an integer number of dollars. She paid with a $50 bill. The cashier gave her an amount of change where the digits of the change, when added together, sum up to a prime number. Also, the change was an even number. If the item costs more than $1, which of the following *must* be true about the number of items Maya bought?
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Solution
If the number of items is an odd number, then the cost of each item must be an even number. — Let 'I' be the cost of one item and 'N' be the number of items. The total cost is N*I. Let 'C' be the change received.
From the problem statement:
1. Maya paid with $50, so N*I + C = 50.
2. C is an even number.
3. The sum of the digits of C is a prime number.
4. I is an integer and I > $1.
First, let's analyze the relationship between N*I and C. Since C is an even number, 50 - C must also be an even number. Therefore, N*I must always be an even number.
Now, let's consider the implications of N*I being an even number:
- For a product of two integers (N*I) to be even, at least one of the integers (N or I) must be even.
Let's evaluate the given choices:
A. 'The number of items is always a composite number.'
If N=2 (e.g., N*I=38, I=19), 2 is a prime number. So, this statement is false.
B. 'The number of items is always an even number.'
If N=3 (e.g., N*I=12, I=4), 3 is an odd number. So, this statement is false.
C. 'If the number of items is an odd number, then the cost of each item must be an even number.'
As established, N*I must be an even number. If N is an odd number, for N*I to be even, I *must* be an even number. This statement holds true in all valid scenarios.
D. 'The number of items is always a factor of 48.'
Possible values for N*I are 48, 38, 36, 34, 30, 20, 18, 16, 12 (derived from possible even 'C' values whose digits sum to a prime: C=2 (48), C=12 (38), C=14 (36), C=16 (34), C=20 (30), C=30 (20), C=32 (18), C=34 (16), C=38 (12)).
If N*I = 38, N can be 1, 2, 19 (since I>1). N=19 is not a factor of 48. So, this statement is false.
Therefore, the only statement that *must* be true is choice C.
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