If
n
is a two - digit number whose units digit is
4
times its tens digit, which of the following statements must be true?
A.
n
is a multiple of
4
.
B.
n
is a multiple of
7
.
C.
10
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Let's solve this step by step. We have a two-digit number where the units digit is 4 times the tens digit. Let's represent this number as n equals 10t plus u, where t is the tens digit and u is the units digit. The given condition is u equals 4t.
Now let's find which values are valid. For a two-digit number, the tens digit t must be between 1 and 9, and the units digit u must be between 0 and 9. Since u equals 4t, we need 0 ≤ 4t ≤ 9. Let's check each possibility: If t equals 1, then u equals 4, giving us n equals 14. If t equals 2, then u equals 8, giving us n equals 28. If t equals 3 or more, then u would be 12 or greater, which is not a valid single digit. So the only possible values are 14 and 28.
Now let's test each statement against both possible values: 14 and 28. Statement A says n is a multiple of 4. For 14: 14 divided by 4 equals 3.5, so false. For 28: 28 divided by 4 equals 7, so true. Since it's not true for both, A is incorrect. Statement B says n is a multiple of 7. For 14: 14 divided by 7 equals 2, so true. For 28: 28 divided by 7 equals 4, so true. This works for both values! Let me quickly check the others: C, D, and E all fail for at least one value. Therefore, the answer is B.
Let me show you why statement B must always be true. We know that n equals 10t plus u, and since u equals 4t, we can substitute to get n equals 10t plus 4t, which simplifies to 14t. Since 14 equals 2 times 7, we can write n as 7 times 2t. This proves that n is always a multiple of 7, regardless of which valid value of t we choose. We can verify this: when t equals 1, n equals 14 which is 7 times 2. When t equals 2, n equals 28 which is 7 times 4. The answer is definitively B.
To summarize our solution: We determined that n can only be 14 or 28, since these are the only two-digit numbers where the units digit is 4 times the tens digit. We then tested all five statements and found that only statement B is true for both values. We proved this algebraically by showing that n equals 14t, which is always a multiple of 7. Therefore, the correct answer is B: n is a multiple of 7.